A General Precipitation-limited L_X–T–R Relation among Early-type Galaxies

Massive galactic systems have long been known to have X-ray luminosities (${L}_{{\rm{X}}}$) that are closely related to the temperature (T) of the diffuse ambient gas that fills them (Sarazin 1986; Mulchaey 2000; Mathews & Brighenti 2003). In clusters of galaxies (${kT}\gtrsim 2$ keV), the observed relation is ${L}_{{\rm{X}}}\propto {T}^{\zeta }$ with $\zeta \approx 2.5\mbox{--}3$ (e.g., Mushotzky 1984; Markevitch 1998; Pratt et al. 2009; Maughan et al. 2012; Giles et al. 2016). Groups of galaxies (1 keV ≲ kT ≲ 2 keV) exhibit a slightly steeper power-law relationship with ζ ≈ 3–3.5 (e.g., Osmond & Ponman 2004; Sun et al. 2009). Among early-type galaxies with ${kT}\lesssim 1\,\mathrm{keV}$, the observed relationship steepens further to $\zeta \approx 4.5$ (e.g., O'Sullivan et al. 2003; Boroson et al. 2011; Kim & Fabbiano 2015; Goulding et al. 2016).

The ${L}_{{\rm{X}}}$–T relationship steepens toward lower temperatures because nongravitational processes, such as radiative cooling and energetic feedback, have progressively greater effects on the structure of the ambient medium as the depth of the confining gravitational potential declines (e.g., Evrard & Henry 1991; Kaiser 1991; Ponman et al. 1999; Voit 2005). In the absence of cooling and feedback, gravitational structure formation would have produced a nearly self-similar family of objects with ${L}_{{\rm{X}}}\propto {T}^{3/2}{\rm{\Lambda }}(T)$, where Λ is the usual cooling function for optically thin gas in collisional ionization equilibrium. However, ambient gas in a galactic system radiates an amount of energy similar to its thermal energy in a time ${t}_{\mathrm{cool}}=3{nkT}/2{n}_{e}{n}_{i}{\rm{\Lambda }}(T)$, where n_e, n_i, and n are the electron density, ion density, and total number density, respectively. Near the center of a typical galactic system, this cooling time is less than the age of the universe. Consequently, radiative cooling and whatever feedback it triggers inevitably modify the density distribution of the ambient medium and the resulting ${L}_{{\rm{X}}}$–T relation, with more pronounced effects at lower temperatures (e.g., Voit & Bryan 2001; Voit et al. 2002).

Numerical simulations of enormous sophistication and complexity have been devoted to modeling the interplay between cooling and feedback in massive galaxies (e.g., Hopkins et al. 2014; Vogelsberger et al. 2014; Schaye et al. 2015), but observations are increasingly indicating that a phenomenological principle of surprising simplicity emerges from those complex interactions: feedback triggered by the production of cold gas clouds appears to limit ${t}_{\mathrm{cool}}$ at a given radius R to be no less than $\approx 10$ times the freefall time ${t}_{\mathrm{ff}}={(2R/g)}^{1/2}$, where g is the gravitational acceleration at R (e.g., Sharma et al. 2012a). This lower limit on ${t}_{\mathrm{cool}}(R)$ is observed among both central cluster galaxies (Voit et al. 2015a; Hogan et al. 2017) and other massive elliptical galaxies (Voit et al. 2015b). It is also observed in numerical simulations in which chaotic accretion of cold gas clouds toward the central black hole fuels strong bipolar outflows (e.g., Gaspari et al. 2012; Li et al. 2015; Meece et al. 2017).

Initially, the theoretical rationale for a feedback-enforced lower limit at ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\approx 10$ was framed in terms of thermal instability (McCourt et al. 2012; Sharma et al. 2012b). However, numerical simulations have since shown that uplift of low-entropy ambient gas is at least as important to production of cold clouds. As in a thunderstorm, the adiabatic cooling that occurs during uplift induces a phase transition (e.g., Revaz et al. 2008; Li & Bryan 2014; McNamara et al. 2016). In this case, uplift induces condensation of gas clouds much cooler and denser than the ambient gas. Modest amounts of vertical mixing, therefore, lead to the development of a multiphase structure in gas that initially has ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\lesssim 10$. Newly formed cold clouds then rain toward the center of the galaxy (e.g., Gaspari et al. 2013, 2017; Gaspari & Sadowski 2017), and the analogy with terrestrial weather has led us to call the overall process "precipitation." It is self-regulating because feedback fueled by the condensates heats the ambient medium until $\min ({t}_{\mathrm{cool}}/{t}_{\mathrm{ff}})\gtrsim 10$, and that rise in ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ inhibits further precipitation, for reasons discussed extensively in Voit et al. (2017).

This paper presents evidence showing that the "precipitation limit" observed at ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\approx 10$ in massive galactic systems extends down to Milky Way scales. In Section 2, we compute the maximum X-ray luminosity that can come from within radius R in a system with ${t}_{\mathrm{cool}}\gtrsim 10{t}_{\mathrm{ff}}$ at all radii. In Section 3, we compare this limit to observations of massive galactic systems ranging from galaxy clusters down to individual early-type galaxies from the ATLAS${}^{3{\rm{D}}}$ (Cappellari et al. 2011) and MASSIVE (Ma et al. 2014) samples. This comparison shows that the upper envelope of L_X within R among these systems matches the precipitation limit over nearly 7 orders of magnitude. Section 4 then builds the precipitation limit into a more general model for the L_X–T–R relation and compares it with some representative data. Section 5 summarizes our results.

In a spherically symmetric system, the X-ray luminosity from within radius R is

$$\begin{eqnarray}&&{L}_{{\rm{X}}}(R)={\int }_{0}^{R}4\pi {r}^{2}\,{n}_{e}{n}_{i}{\rm{\Lambda }}(T)\,{dr}.\end{eqnarray} \tag{ 1 }$$

According to the precipitation framework, feedback triggered by condensation enforces ${t}_{\mathrm{cool}}\gtrsim 10{t}_{\mathrm{ff}}$ and places an upper bound on the electron density of the ambient gas:

$$\begin{eqnarray}&&{n}_{e}\lesssim \displaystyle \frac{3{kT}}{10\,{t}_{\mathrm{ff}}{\rm{\Lambda }}(T)}\left(\displaystyle \frac{n}{2{n}_{i}}\right).\end{eqnarray} \tag{ 2 }$$

Assuming an isothermal potential (${t}_{\mathrm{ff}}\propto R$) in which the ambient gas temperature remains approximately constant with radius, we obtain from this constraint the following upper limit on the X-ray luminosity coming from within R:

$$\begin{eqnarray}&&{L}_{{\rm{X}}}(R)\lesssim \displaystyle \frac{9\pi }{25}\displaystyle \frac{{({kT})}^{2}}{{\rm{\Lambda }}(T)}\,{\sigma }_{v}^{2}\,R.\end{eqnarray} \tag{ 3 }$$

Here, ${\sigma }_{v}={({gR})}^{1/2}$ is the line-of-sight component of an isotropic velocity dispersion corresponding to the potential-well depth, and we have dropped the insignificant ${n}^{2}/4{n}_{e}{n}_{i}$ factor for simplicity.

Expressing ${\sigma }_{v}$ in terms of the parameter $\beta \equiv \mu {m}_{p}{\sigma }_{v}^{2}/{kT}$ reduces this upper limit on X-ray luminosity to a function of T and R:

$$\begin{eqnarray}&&{L}_{{\rm{X}}}(R)\lesssim \displaystyle \frac{9\pi }{25}\displaystyle \frac{\beta }{\mu {m}_{p}}\displaystyle \frac{{({kT})}^{3}}{{\rm{\Lambda }}(T)}\,R.\end{eqnarray} \tag{ 4 }$$

In systems that are close to hydrostatic equilibrium, the value of β reflects the power-law slope of the radial gas pressure gradient, because $d\mathrm{ln}P/d\mathrm{ln}r\approx -2\beta$. A nearly isothermal and hydrostatic system at the precipitation limit therefore has $P(r)\mathop{\propto }\limits_{\sim }{n}_{e}(r)\mathop{\propto }\limits_{\sim }{r}^{-1}$ and $\beta \approx 0.5$.

The factor of ${\rm{\Lambda }}(T)$ in the denominators of Equations (2)–(4) is a novel feature of precipitation-limited systems. It ends up in the denominator because raising ${\rm{\Lambda }}(T)$ increases the ambient medium's capacity for cooling and condensation. The resulting energetic feedback therefore does not diminish until it drives down the ambient gas density far enough to limit condensation. Within a fixed radius R, the limiting L_X–T relation in the free–free cooling regime (${kT}\gtrsim 2$ keV) is characterized by $\zeta \approx 2.5$. At lower temperatures, the relation becomes steeper, as emission-line cooling starts to dominate. These features agree well with observations of the L_X–T relation and provide a particularly natural explanation for the observed steepening below 2 keV. However, a proper comparison between theory and observation becomes more difficult at lower temperatures because cooler systems have substantially lower X-ray surface brightnesses. This feature complicates the task of measuring L_X(R) at fixed R over a broad range in system mass. So, instead, we will proceed to compare observations of L_X(R) to the predicted luminosity limit over the ranges in R for which good measurements are available.

Our comparison in Figure 1 of observations with the limiting bolometric X-ray luminosity predicted by Equation (3) draws on several different data sets:

• 1.  
  Galaxy-cluster cores. The ACCEPT database (Cavagnolo et al. 2009) is our source for the bolometric X-ray luminosities and temperatures of galaxy-cluster cores. Those luminosities were originally measured in the 0.7–7.0 keV band, and we determine bolometric luminosities within R by integrating over deprojected density and temperature profiles from ACCEPT. We use the subset analyzed by Voit et al. (2015b), which separates into two categories: (1) multiphase clusters with detectable Hα emission (blue lines with triangles) and (2) single-phase clusters with optical spectroscopy showing no detectable Hα emission (red lines with triangles). In order to compute the precipitation limit on L_X(R), we use an approximation to ${\sigma }_{v}(R)$ consisting of the sum of a singular isothermal sphere with ${\sigma }_{v}=300\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and an NFW profile (Navarro et al. 1997) of fixed concentration and an amplitude determined by the cluster's X-ray temperature. Concentration is defined with respect to the radius R₅₀₀ that encloses a mean matter density 500 times the critical density, and the scale radius at which the mass-density profile is ∝ R⁻² is assumed to be ${R}_{s}={R}_{500}/3$. We show tracks of L_X(R) that go from $R\leqslant 10\,\mathrm{kpc}$ through R = 300 kpc, but focus particularly on measurements of L_X(R) within the apertures $0.04\,{R}_{500}$ (filled triangles) and $0.15\,{R}_{500}$ (open triangles). The larger aperture is commonly used to separate the core regions of a cluster from its outer parts. The smaller aperture yields high signal-to-noise measurements of core properties. For a 5 keV galaxy cluster, our model for the gravitational potential gives ${\sigma }_{v}(0.04{R}_{500})=454\,\mathrm{km}\,{{\rm{s}}}^{-1}$, ${\sigma }_{v}(0.15{R}_{500})=630\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and $\max ({\sigma }_{v})=760\,\mathrm{km}\,{{\rm{s}}}^{-1}$.
• 2.  
  Galaxy-group cores. Our primary source for the luminosities and temperatures of galaxy-group cores (teal squares) is Sun et al. (2009). For these objects, we use the luminosity within the radius $0.15{R}_{500}$, and R₅₀₀ is obtained from a hydrostatic mass model for each group. Surface brightness was originally measured in the 0.7–2 keV band and the bolometric correction is from PIMMS. We do not have direct measurements of ${\sigma }_{v}(0.15{R}_{500})$ and so instead we use Equation (4) with $\beta =0.5$ to obtain precipitation limits on L_X(R). Our secondary source is the Panagoulia et al. (2014) sample of groups and clusters. That compilation contains 13 systems with ${kT}\approx 1\,\mathrm{keV}$ that have a mean entropy profile $K{(R)=95.4\mathrm{keV}{\mathrm{cm}}^{2}\times (R/100\mathrm{kpc})}^{2/3}$, where $K\equiv {{kTn}}_{e}^{-2/3}$. Representative values of L_X(R) and precipitation limits can be derived for these systems by setting ${kT}=1\,\mathrm{keV}$ and applying Equation (4) with $\beta =0.5$.
• 3.  
  Werner ellipticals. High-quality Chandra observations of 10 nearby massive elliptical galaxies were presented by Werner et al. (2012, 2014) and analyzed by Voit et al. (2015b) in the context of precipitation-regulated feedback. Surface brightness was originally measured in the 0.7–7.0 keV band, and we determine bolometric luminosities within R by integrating over deprojected density and temperature profiles. Five of them (blue lines with diamonds) contain multiphase gas that extends over several kiloparsecs. The other five (mostly red lines with hexagons) are considered single-phase galaxies because there is no observable multiphase gas beyond the central kiloparsec. However, one of them (NGC 4261, purple line with stars) contains a central subkiloparsec disk of cool, dusty gas that presumably fuels the galaxy's strong bipolar outflow, which is roughly two orders of magnitude more powerful than the AGN outflows in the other nine galaxies. Deprojected density and temperature profiles from those studies are used to reconstruct L_X(R). Precipitation limits on L_X(R) are computed for those galaxies using ${\sigma }_{v}$ values from the Hyperleda database⁶ under the assumption that ${\sigma }_{v}(R)$ is constant. The lines go from $R\lt 1\,\mathrm{kpc}$ to $R\gt 10\,\mathrm{kpc}$, with endpoints limited by data quality. Filled symbols mark R = 1 kpc and open ones indicate R = 5 kpc.
• 4.  
  Volume-limited surveys. Ambient gas in early-type galaxies of lower mass is harder to observe because of its lower surface brightness. The radii out to which diffuse X-ray emission can be reliably measured are correspondingly smaller, and Chandra observations are required in order to excise the X-ray emission from individual X-ray binaries. Here, we take advantage of a Chandra archival analysis by Goulding et al. (2016) of 74 early-type galaxies from two volume-limited samples: ATLAS${}^{3{\rm{D}}}$ (${M}_{* }\gt {10}^{9.9}\,{M}_{\odot }$ and within 42 Mpc) and MASSIVE (${M}_{* }\gt {10}^{11.5}\,{M}_{\odot }$ and within 108 Mpc). Measurements of each galaxy's bolometric⁷ X-ray luminosity within an aperture corresponding to the effective radius (R_e) of starlight are shown with orange squares (ATLAS${}^{3{\rm{D}}}$) and green circles (MASSIVE). Some of those galaxies also contain detectable multiphase gas. Overplotted pink dots indicate galaxies with CO detections from Davis et al. (2011, 2016). Cyan symbols indicate galaxies with detectable nebular optical line emission (Pandya et al. 2017). Among the cyan symbols, triangles represent extended emission, while circles represent unresolved detections.
• 5.  
  Milky Way. Our L_X(R) tracks for the Milky Way (green lines with stars) are derived from the models of Miller & Bregman (2015), which represent fits to Chandra observations of O vii (dotted line and open star) and O viii (solid line and filled star) emission lines along many different lines of sight through the Galaxy. We use the best fitting parameters in their optically thin models. The tracks go from R = 1 to 10 kpc, and stars indicate R = 8 kpc.

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3.1. The Precipitation Limit

3.2. The ${L}_{X}({R}_{e})$–T Relation

Figure 2 shows the luminosity–temperature relation that results when L_X(R) is divided by the aperture radius R in order to remove the aperture dependence of the precipitation limit. Most of the galaxy points correspond to the aperture R_e. The cluster-core points correspond to $0.04{R}_{500}$, which is typically ∼40 kpc.⁸ A short-dashed purple line shows the ${L}_{X}({R}_{e})$–T fit from Goulding et al. (2016), divided by the R_e(T) fit from the previous paragraph. Gray lines show tracks derived from a version of Equation (4) that has been generalized to make ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ a free parameter and in which $\beta =0.5$ to accord with the power-law slope of the precipitation-limited pressure profile:

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{X}(R)}{R}=\displaystyle \frac{18\pi }{\mu {m}_{p}}\displaystyle \frac{{({kT})}^{3}}{{\rm{\Lambda }}(T)}{\left(\displaystyle \frac{{t}_{\mathrm{cool}}}{{t}_{\mathrm{ff}}}\right)}^{-2}.\end{eqnarray} \tag{ 5 }$$

Making this generalization allows the figure to show how the data points reflect the typical value of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ corresponding to a particular combination of ${L}_{X}(R)\cdot {R}^{-1}$ and kT. For comparison, a teal dotted line shows

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{X}({R}_{e})}{{R}_{e}}=\displaystyle \frac{12\pi {R}_{e}^{2}}{{(1\mathrm{Gyr})}^{2}}\displaystyle \frac{{({kT})}^{2}}{{\rm{\Lambda }}(T)},\end{eqnarray} \tag{ 6 }$$

for ${R}_{e}\approx (8.8\,\mathrm{kpc}){({kT}/1\mathrm{keV})}^{0.9}$, which describes systems with a constant cooling time ${t}_{\mathrm{cool}}=1\,\mathrm{Gyr}$ inside of R_e.

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The track corresponding to ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}=10$ again follows the upper envelope of the data points but now changes in slope because of the ${\rm{\Lambda }}(T)$ factor in Equation (6). Most of the data points follow the same trend with T but correspond to greater ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ values, centered near ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\approx 25$. This finding supports the hypothesis that the change in L_X–T slope observed near ∼1 keV among early-type galaxies is caused by precipitation-regulated feedback. While ratios as great as ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\gtrsim 25$ may seem too large for condensation and precipitation, we would like to emphasize that the MASSIVE and ATLAS${}^{3{\rm{D}}}$ points indicate average values of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ within R_e. The minimum values of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ may be considerably smaller, especially near the center. For examples, see the tracks belonging to the Werner ellipticals in Figure 1, which are close to the precipitation limit at ∼1 kpc but can diverge from it at larger radii.

Both Kim & Fabbiano (2015) and Goulding et al. (2016) point out that recent simulations by Choi et al. (2015) obtain a similar L_X–T relation for early-type galaxies with ${kT}\lesssim 1\,\mathrm{keV}$ when the AGN feedback is kinetic instead of thermal. This change in feedback mode reduces L_X by two orders of magnitude in their simulations, but why does that happen? Meece et al. (2017) have shown that changing the mode of AGN energy injection from thermal to kinetic has deep implications for precipitation and self-regulation. Pure thermal energy injection at a galaxy's center fails to bring about self regulated AGN feedback because it overturns the surrounding entropy gradient. The resulting convection promotes thermal instability and triggers runaway condensation.

Voit et al. (2017) discuss these issues in detail and show that self-regulation requires an energy-injection mechanism that deposits heat into the circumgalactic medium without inverting its entropy gradient. Bipolar AGN jets are one such energy injection mechanism, but the key feature is not the kinetic energy. Instead, it is the response of the global entropy gradient to energy input. If feedback maintains a rising entropy profile outside the central few kiloparsecs, then self-regulation through precipitation will naturally produce the observed L_X–T relation.

Models that assume a universal power-law entropy profile are less successful. For example, Panagoulia et al. (2014) find that the mean profile $K{(R)=95.4\mathrm{keV}{\mathrm{cm}}^{2}\times (R/100\mathrm{kpc})}^{2/3}$ derived from systems with ${kT}\approx 1\,\mathrm{keV}$ is also an adequate description of many galaxy-cluster cores. However, the relation derived from a universal entropy profile with this slope scales as ${L}_{X}(R)\propto {T}^{3}{\rm{\Lambda }}(T)R$, as shown by the long-dashed line in Figure 2. Its slope is similar to the precipitation limit in the ${kT}\approx 0.3\mbox{--}1.0$ keV range but diverges from the precipitation slope at both higher and lower temperatures. Figure 2 also demonstrates that the Panagoulia et al. (2014) profile cannot be universal, because of the large dispersion in ${L}_{X}(R)\cdot {R}^{-1}$ at each temperature. The cluster-core points in particular span more than three orders of magnitude in ${L}_{X}(R)\cdot {R}^{-1}$, which requires the core entropy of the ambient medium to span at least an order-of-magnitude range in objects of a given temperature.

3.3. Incidence of Multiphase Gas

Our objective in this section is to present a maximally simple model for the overall L_X–T relation that incorporates the precipitation limit (see also Sharma et al. 2012a). The standard aperture for this relation among groups and clusters of galaxies is R₅₀₀. X-ray surface-brightness profiles of massive galaxy clusters are frequently well-observed out to at least this radius (e.g., Maughan et al. 2012). Galaxy groups can also be observed out to this radius but require a careful treatment of the soft X-ray backgrounds (e.g., Sun et al. 2009). X-ray emission from individual galaxies cannot be measured out to R₅₀₀, but a mean ${L}_{X}({R}_{500})$–T relation for massive galaxies can be obtained by stacking ROSAT observations of large numbers of galaxies (Anderson et al. 2015).⁹ Figure 3 shows a representative selection of such measurements, with no corrections made to account for self-similar cosmological evolution. For comparison, the figure also shows the L_X–T points from Figures 1 and 2, which represent measurements within smaller apertures. Those points show that the slope of the observed L_X–T relation at $\lesssim 2\,\mathrm{keV}$ is aperture dependent. Within the aperture R₅₀₀, the slope of the L_X–T relation remains nearly constant over the entire observed temperature range.

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The gray shading in Figure 3 shows a model for the ${L}_{X}({R}_{500})$–T relation that combines a cosmological entropy profile at large radii with a precipitation-limited entropy profile at small radii. This model accounts for the lack of a break in slope. The cosmological entropy profile is extremely simple. It assumes that the gravitational potential is a singular isothermal sphere in which the baryon mass fraction at each radius is equal to the cosmological baryon fraction. In hydrostatic equilibrium, the gas temperature is ${kT}=\mu {m}_{p}{\sigma }_{v}^{2}$, giving an entropy profile

$$\begin{eqnarray}&&{K}_{\cos }(R)=\mu {m}_{p}{\sigma }_{v}^{2}{\left[\displaystyle \frac{125{f}_{b}{H}^{2}(z)}{2\pi G{\mu }_{e}{m}_{p}}\right]}^{-2/3}{\left(\displaystyle \frac{R}{{R}_{500}}\right)}^{4/3},\end{eqnarray} \tag{ 7 }$$

where H(z) is the Hubble expansion parameter and ${\mu }_{e}$ is the mean mass per electron. Without modification, gas with this entropy profile would produce an X-ray luminosity that diverges at small radii. In non-radiative cosmological simulations, mixing produces an entropy core that keeps the X-ray luminosity from diverging (e.g., Mitchell et al. 2009), but when radiative cooling is turned on, L_X is limited by a combination of cooling and feedback (e.g., Nagai et al. 2007).

In the precipitation framework, the limiting entropy profile at small radii depends on the ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ ratio at which condensation-triggered feedback self-regulates. Consequently, the X-ray luminosity of a precipitation-regulated system is determined by the limiting value of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$. The electron density profile corresponding to this limit is given by a generalization of Equation (2):

$$\begin{eqnarray}&&{n}_{e,\mathrm{pre}}(R)=\displaystyle \frac{3{kT}{\sigma }_{v}}{{\rm{\Lambda }}(T)}\left(\displaystyle \frac{n}{2{n}_{i}}\right){\left(\displaystyle \frac{{t}_{\mathrm{cool}}}{{t}_{\mathrm{ff}}}\right)}^{-1}{R}^{-1}.\end{eqnarray} \tag{ 8 }$$

The temperature of hydrostatic, isothermal gas with this density profile is ${kT}=2\mu {m}_{p}{\sigma }_{v}^{2}$ in an isothermal potential and gives the limiting entropy profile

$$\begin{eqnarray}&&{K}_{\mathrm{pre}}(R)={(2\mu {m}_{p})}^{1/3}{\left[\displaystyle \frac{2{n}_{i}{\rm{\Lambda }}(T)}{3n}\right]}^{2/3}{\left(\displaystyle \frac{{t}_{\mathrm{cool}}}{{t}_{\mathrm{ff}}}\right)}^{2/3}{R}^{2/3}.\end{eqnarray} \tag{ 9 }$$

When calculating ${K}_{\mathrm{pre}}(r)$, we use a cooling function with solar metallicity, because AGN triggering in a massive galaxy happens in regions where the gas-phase abundances are approximately solar. The X-ray luminosity of gas with this entropy profile diverges toward large radii and is ultimately limited by the cosmological entropy profile.

Adding the generalized precipitation profile to the cosmological profile gives a combined entropy profile that depends only on ${\sigma }_{v}$ and the limiting value of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$:

$$\begin{eqnarray}&&K(R)={K}_{\cos }(R)+{K}_{\mathrm{pre}}(R).\end{eqnarray} \tag{ 10 }$$

The power-law slope of this combined profile is $K\propto {R}^{2/3}$ at small radii and asymptotically approaches $K\propto {R}^{4/3}$ at large radii. In hydrostatic equilibrium in an isothermal potential well, the pressure profile of a nearly isothermal gas is $d\mathrm{ln}P/d\mathrm{ln}r\approx -(3/2)(d\mathrm{ln}K/d\mathrm{ln}R)$. One can therefore obtain an approximate temperature profile

$$\begin{eqnarray}&&{kT}(R)=\displaystyle \frac{\mu {m}_{p}{\sigma }_{v}^{2}\ K(R)}{{K}_{\cos }(R)+0.5{K}_{\mathrm{pre}}(R)}\end{eqnarray} \tag{ 11 }$$

from the logarithmic derivative of K(R), which leads to the approximate electron density profile

$$\begin{eqnarray}&&{n}_{e}(R)={\left[\displaystyle \frac{{K}_{\cos }(R)+0.5{K}_{\mathrm{pre}}(R)}{\mu {m}_{p}{\sigma }_{v}^{2}}\right]}^{-3/2}.\end{eqnarray} \tag{ 12 }$$

Integrating Equation (2) with this density profile then gives the model value of ${L}_{X}({R}_{500})$. The model value of T is simply the emissivity-weighted average of T(R). In both of these integrations, the cooling function ${\rm{\Lambda }}(T)$ depends strongly on metallicity for ${kT}\lesssim 2\,\mathrm{keV}$. We therefore assume a representative metallicity profile $Z{(R)/{Z}_{\odot }=\min [1.0,0.3(R/{R}_{500})}^{-1/2}]$ that is broadly consistent with measurements of galaxy groups (e.g., Rasmussen & Ponman 2007; Sun et al. 2009).

Gray lines in Figure 3 are ${L}_{X}({R}_{500})$–T tracks corresponding to different values of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ in Equation (8). This ratio is the only free parameter in the model and specifies the asymptotic value of ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}$ at small radii. Setting ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}=10$ gives a track that follows the upper envelope of the observations from $\approx 10\,\mathrm{keV}$ down through $\lesssim 1\,\mathrm{keV}$, at which X-ray measurements extending out to R₅₀₀ in individual objects become extremely difficult. That track changes slope in two places. A subtle change happens near 1 keV. The mean value of $\beta =\mu {m}_{p}{\sigma }_{v}^{2}/{kT}$ rises from $\approx 0.5$ to $\approx 1$ as kT increases through ∼1 keV. This change in β shifts points with ${kT}\lesssim 1\,\mathrm{keV}$ to greater temperatures for a given ${\sigma }_{v}$, which causes the ${L}_{X}({R}_{500})$–T relation to steepen near 1 keV. A more pronounced change in slope happens near 0.2 keV, below which the entire profile out to R₅₀₀ is essentially the same as the precipitation-limited profile. In that regime, which includes the Milky Way, the model predicts ${L}_{X}({R}_{500})\propto {T}^{3.5}{{\rm{\Lambda }}}^{-1}(T)$, with ${\rm{\Lambda }}\mathop{\propto }\limits_{\sim }{T}^{-1}$.

Each of the data sets in Figure 3 suffers from biases that we will not fully analyze here. For example, X-ray selected surveys of clusters and groups of galaxies tend to be biased toward the more X-ray luminous examples in each mass and temperature range and overestimate the mean ${L}_{X}({R}_{500})$ at a given T if that bias is not properly corrected (e.g., Allen et al. 2011). Conversely, optically selected samples can tend to underestimate ${L}_{X}({R}_{500})$ at a given T because of how scatter in the relation between the optical mass proxy and the actual system mass combines with the steep slope of the mass function (e.g., Rozo et al. 2014). A blend of these two biases is responsible between the offset between the ${L}_{X}({R}_{500})$–T relations derived from the X-ray selected samples (REXCESS and XXL) and the ${L}_{X}({R}_{500})$–T relation derived by Anderson et al. (2015) from ROSAT stacks of luminous red galaxies from the Sloan Digital Sky Survey. An additional contribution to the offset may come from anticorrelation of L_X with galactic stellar mass at fixed total mass.

However, despite those biases, the slopes of the ${L}_{X}({R}_{500})$–T relations derived from REXCESS and XXL remain similar to the slope of the Anderson et al. (2015) points down to temperatures at least a factor of 2 lower than the bottom end of the range covered by the X-ray surveys, and maybe even down to the Milky Way itself. Furthermore, the ${L}_{X}({R}_{500})$–T relation derived from our simple model shares the same slope. Taken together, these results strongly suggest that feedback regulation of circumgalactic gas around galactic systems with ${kT}\approx 0.2\mbox{--}1.0$ keV shares much in common with the AGN feedback mechanism that regulates the cores of massive galaxy clusters.

In galaxy clusters, strong AGN feedback is tightly linked to the presence of a cool core, which in turn is linked with the presence of multiphase gas in the central galaxy (McNamara & Nulsen 2007, 2012; Cavagnolo et al. 2008). Maughan et al. (2012) have measured separate ${L}_{X}({R}_{500})$–T relations for cool-core and non-cool-core clusters, as well as ${L}_{X}({R}_{500})$–T relations excluding the core region within $0.15{R}_{500}$. When the cores are excluded, the ${L}_{X}({R}_{500})$–T relations for cool-core ($\propto {T}^{2.1}$) and non-cool-core clusters ($\propto {T}^{2.9}$) overlap one another, with a slope $\propto {T}^{2.7}$ for the combined sample. These populations overlap because the radial profiles of entropy and electron density beyond $0.15{R}_{500}$ are nearly identical to the cosmological profile for clusters with ${kT}\gtrsim 3.5\,\mathrm{keV}$. However, the core regions of those two cluster populations are quite different.

Figure 3 shows the ${L}_{X}(0.15{R}_{500})$–T relations we obtain from Maughan et al. (2012) by subtracting the core-excluded relations from the full-aperture relations. The lines for cool-core and non-cool-core clusters may be compared with the ACCEPT data points for multiphase and single-phase clusters, respectively, because the root populations are very similar. Cool-core clusters tend to reside near the precipitation limit, with an L_X–T slope similar to the model prediction, while the non-cool-core clusters reside below it, with a steeper L_X–T slope. Apparently, the processes that shut off production of multiphase gas have a greater impact on core structure in galaxy clusters with shallower potential wells.

Galaxy clusters with central cooling time $\gt 1\,\mathrm{Gyr}$ generally do not contain multiphase gas in their cores, and this trend appears to extend down through the ATLAS${}^{3{\rm{D}}}$ galaxies. A dotted teal line in Figure 3 shows the ${L}_{X}({R}_{e})$–T relation corresponding to ambient gas with a constant cooling time ${t}_{\mathrm{cool}}=1\,\mathrm{Gyr}$ within ${R}_{e}={(8.8\mathrm{kpc})({kT}/1\mathrm{kpc})}^{0.9}$. Systems below this line must have central cooling times exceeding 1 Gyr, and the line runs close to the transition from single-phase to multiphase ACCEPT cluster cores. At lower temperatures, all of the multiphase systems lie above this line, but some of the single-phase systems lie below it. This pattern may represent another link between the feedback mechanism in galaxy clusters and the mechanisms that regulate early-type galaxies of lower mass.

This paper has compiled a broad array of L_X–T data measured within a variety of apertures and has compared it with a simple model based on the precipitation framework for AGN feedback. Its main findings are as follows:

• 1.  
  The maximum measured values of L_X(R) track the theoretical precipitation limit derived from the condition ${t}_{\mathrm{cool}}\gtrsim 10{t}_{\mathrm{ff}}$ over nearly seven orders of magnitude in L_X, from the cores of massive galaxy clusters all the way down to the Milky Way. This finding strongly suggests that a common mechanism regulates cooling of circumgalactic gas and possibly also quenching of star formation in all of these massive galactic systems.
• 2.  
  When measured within the stellar envelopes of early-type galaxies, the L_X–T relation changes slope from $\mathop{\propto }\limits_{\sim }{T}^{4.5}$ at ${kT}\lesssim 1\,\mathrm{keV}$ to $\mathop{\propto }\limits_{\sim }{T}^{3}$ at ${kT}\gtrsim 2\,\mathrm{keV}$. The change in slope accords with the predictions of precipitation-regulated feedback models, in which ${L}_{X}(R)\propto {T}^{3}{{\rm{\Lambda }}}^{-1}(T)R$. In this context, the bend observed at ∼1 keV is an inversion of the bend in ${\rm{\Lambda }}(T)$ in this same temperature range.
• 3.  
  The L_X–T relation measured within the cosmological aperture R₅₀₀ shows little change in slope over the entire observed temperature range. We construct a model for the ${L}_{X}({R}_{500})$–T relation by combining a precipitation-regulated entropy profile at small radii with a cosmological entropy profile at large radii and show that the resulting relation also shows little change in slope.

Observations of X-ray luminosity out to ∼R₅₀₀ around individual massive galaxies may become possible with the next generation of X-ray telescopes and will have much to teach us about the astrophysics of AGN feedback and star formation quenching. The large gap between our model for ${L}_{X}({R}_{500})$–T and measurements of ${L}_{X}({R}_{e})$ in the 0.2–0.7 keV temperature range suggests that such observations will be fruitful. In this context, it is notable that both the Anderson et al. (2015) points and the Milky Way points at 100 kpc are consistent with our ${L}_{X}({R}_{500})$–T model for a median ${t}_{\mathrm{cool}}/{t}_{\mathrm{ff}}\approx 20\mbox{--}30$, since this is the same median that best describes the L_X(R)–T relation measured within the stellar envelopes of early-type galaxies containing multiphase gas. It is therefore likely that the X-ray luminosity from within R₅₀₀ around at least some of the MASSIVE and ATLAS^3D galaxies is at least an order of magnitude greater than the luminosity from within R_e. Measurements by Goulding et al. (2016) of L_X within larger surface-brightness limited apertures do indeed show that ${L}_{X}({R}_{500})\gg {L}_{X}({R}_{e})$ for some of their galaxies but also suggest large scatter in the ${L}_{X}({R}_{500})/{L}_{X}({R}_{e})$ ratio. Separating the aperture effects from intrinsic scatter in ${L}_{X}({R}_{500})$ will require a more detailed analysis, but quite possibly, the results of AGN feedback might turn out to be much more systematic and predictable than one would guess from watching movies of numerical simulations.

We would like to thank Joel Bregman, Greg Bryan, Ali Crocker, Gus Evrard, Mike McCourt, Brian McNamara, Brian O'Shea, and Prateek Sharma for helpful conversations. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915, because G.M.V. benefitted from the Kavli Institute for Theoretical Physics while participating in their workshop on the Galaxy-Halo Connection. C.P.M. acknowledges grants NSF AST-1411945 and 1411642. J.G. also acknowledges NSF AST-1411642. M.S. would like to thank NASA for support through Chandra Award Number AR7-18016X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.