Massive Warm/Hot Galaxy Coronae. II. Isentropic Model

Since the Galactic corona may be heated by active galactic nucleus (AGN) feedback and star formation, we imagine that it evolves toward a convective equilibrium. We therefore adopt an adiabatic EoS relating the gas pressure and mass density,

$$\begin{eqnarray}&&P(r)=K\rho {\left(r\right)}^{\gamma },\end{eqnarray} \tag{ 1 }$$

where r is the radius and K is the entropy parameter, which we assume is constant with radius. Using the ideal gas law allows us to relate the temperature to the density,

$$\begin{eqnarray}&&T(r)=K\displaystyle \frac{\bar{m}}{{k}_{{\rm{B}}}}\rho {\left(r\right)}^{\gamma -1},\end{eqnarray} \tag{ 2 }$$

where $\bar{m}$ is the mean mass per particle.

For a mixture of n fluids, we can write the HSE equation as the sum of the pressures for the different components,

$$\begin{eqnarray}&&{dP}=\sum _{i=1}^{n}{{dP}}_{i}=-\rho d\varphi ,\end{eqnarray} \tag{ 3 }$$

where φ is the gravitational potential. We include three pressure components, similar to those in FSM17: (i) thermal, (ii) nonthermal, from cosmic rays (CRs) and magnetic fields, and (iii) turbulent support. We assume that the density of each component is proportional to the total gravitating gas mass density ρ. For the first two components, we use the adiabatic EoS, with ${\gamma }_{1}=5/3$ and ${\gamma }_{2}=4/3$, respectively, and assume that the entropy parameter is constant with radius. For each component, ${{dP}}_{i}={\gamma }_{i}{K}_{i}{\rho }^{{\gamma }_{i}-1}d\rho$. For the turbulent component, we assume a constant velocity scale, ${\sigma }_{\mathrm{turb}}$, as we did in FSM17 and write ${{dP}}_{3}={\sigma }_{\mathrm{turb}}^{2}d\rho$. Equation (3) is then

$$\begin{eqnarray}&&\left({\sigma }_{\mathrm{turb}}^{2}+\sum _{i=1,2}{\gamma }_{i}{K}_{i}{\rho }^{{\gamma }_{i}-1}\right){\rho }^{-1}d\rho =-d\varphi .\end{eqnarray} \tag{ 4 }$$

Integration then gives

$$\begin{eqnarray}&&{\sigma }_{\mathrm{turb}}^{2}\mathrm{ln}\rho (r)+\sum _{i=1,2}\displaystyle \frac{{\gamma }_{i}}{{\gamma }_{i}-1}{K}_{i}\rho {\left(r\right)}^{{\gamma }_{i}-1}={D}_{b}-{\int }_{{r}_{b}}^{r}\displaystyle \frac{{GM}(r){dr}}{{r}^{2}},\end{eqnarray} \tag{ 5 }$$

where r_b is a reference point, which we normally take at the outer boundary, and D_b is an integration constant.

To solve this equation for $\rho (r)$ for a given mass profile M(r), we must specify ${\sigma }_{\mathrm{turb}}$ and K_i. The former is taken from observations of oxygen line velocities and widths (see Tumlinson et al. 2011a, the discussion in FSM17, and Section 2.3 here). For the latter—since in our model, K_i are constant with radius, they can be expressed as functions of the gas properties at the boundary r_b—the temperature, ${T}_{\mathrm{th},b}$ and density, ${\rho }_{b}$. For the thermal component, this is simply

$$\begin{eqnarray}&&{K}_{1}=\displaystyle \frac{{k}_{{\rm{B}}}}{{\bar{m}}^{{\gamma }_{1}}}\displaystyle \frac{{T}_{\mathrm{th},b}}{{n}_{b}^{{\gamma }_{1}-1}},\end{eqnarray} \tag{ 6 }$$

where ${n}_{b}\equiv {\rho }_{b}/\bar{m}$ is the particle volume density. To obtain K₂, we use the α parameter from FSM17, defined as $\alpha \,\equiv ({P}_{\mathrm{th}}+{P}_{\mathrm{nth}})/{P}_{\mathrm{th}}=({T}_{\mathrm{th}}+{T}_{\mathrm{nth}})/{T}_{\mathrm{th}}$. For isothermal conditions, $\alpha$ is constant with radius. In our new model, the relative fractions of pressure support from each component vary with radius, and $\alpha$ is not constant. We define ${\alpha }_{b}\equiv \alpha ({r}_{b})=({T}_{\mathrm{th},b}+{T}_{\mathrm{nth},b})/{T}_{\mathrm{th},b}$, allowing us to write

$$\begin{eqnarray}&&{K}_{2}=\displaystyle \frac{{k}_{{\rm{B}}}}{{\bar{m}}^{{\gamma }_{2}}}\displaystyle \frac{({\alpha }_{b}-1){T}_{\mathrm{th},b}}{{n}_{b}^{{\gamma }_{2}-1}}.\end{eqnarray} \tag{ 7 }$$

Thus, given ${\sigma }_{\mathrm{turb}}$, and for the gas density, temperature, and α at the reference point, we can solve Equations (4) or (5) for the density profile, $\rho (r)$. We can then use the EoS (Equations (1)–(2)) to find the pressure and temperature profiles for each of the corona components and the total pressure profile.

The metal content of the CGM and its distribution are interesting for two reasons. First, the total metal content provides information on the cumulative metal production in the galaxy by star formation (Peeples et al. 2014). Second, observations of the CGM probe the gas properties, such as density and temperature, mainly through absorption and emission of radiation by metal ions (Spitzer 1956; Bregman 2007; Prochaska et al. 2009; Tumlinson et al. 2017). Thus, metals are important as tracers of the gas distribution.

In FSM17, we assumed a uniform metallicity distribution. In a more realistic scenario, the central region of the Galactic halo is expected to be enriched by metals created in supernova (SN) explosions and ejected from the disk by Galactic winds. The outer regions, close to the virial radius, may be dominated by metal-poor gas accreted from the cosmic web, resulting in a decreasing metallicity profile across the corona. Some of the accreted gas may also be pre-enriched. The level and extent of metal enrichment by outflows from the disk and the enrichment of the accreted gas depends on feedback energetics, the star formation history and distribution in the galaxy, and the physics of gas mixing and diffusion in the corona (see Fielding et al. 2018 and Li & Bryan 2020).

The main observational constraints of our model in this work are oxygen absorption measurements, probing the gas phase metallicity. The mass of metals in the CGM locked in solid-state dust grains is an additional component (Peek et al. 2015), and we do not address it here. As we discuss in Section 3, the mass of metals locked in dust is small compared to gas in our fiducial model, and we do not model the dust.

We adopt a metallicity profile given by

$$\begin{eqnarray}&&Z^{\prime} (r)={Z}_{0}^{{\prime} }{\left[1+{\left(\displaystyle \frac{r}{{r}_{Z}}\right)}^{2}\right]}^{-1/2},\end{eqnarray} \tag{ 8 }$$

where ${Z}_{0}^{{\prime} }$ is the Galactic metallicity and r_Z is an adjustable metallicity length scale within which the metallicity is equal to the inner metallicity ${Z}_{0}^{{\prime} }$ and beyond which the metallicity decreases smoothly to the outer boundary of the CGM, which we denote by ${r}_{\mathrm{CGM}}$. The length scale r_Z can be set by estimating the maximal extent of outflows from the disk. Alternatively, we can set the metallicity at ${r}_{\mathrm{CGM}}$, and then the length scale is given by

$$\begin{eqnarray}&&{r}_{Z}={r}_{\mathrm{CGM}}{\left[{\left(\displaystyle \frac{Z^{\prime} ({r}_{\mathrm{CGM}})}{{Z}_{0}^{{\prime} }}\right)}^{2}-1\right]}^{-1/2}.\end{eqnarray} \tag{ 9 }$$

The mean metallicity is given by

$$\begin{eqnarray}&&{\left\langle Z^{\prime} \right\rangle }_{V}=\displaystyle \frac{\bar{m}}{{M}_{\mathrm{CGM}}}{\int }_{{R}_{0}}^{{r}_{\mathrm{CGM}}}Z^{\prime} (r)n(r){dV},\end{eqnarray} \tag{ 10 }$$

where ${M}_{\mathrm{CGM}}$ is the CGM gas mass. The mean metallicity is calculated over the corona volume, from the inner radius, R₀, to ${r}_{\mathrm{CGM}}$. The total mass of metals in the corona is then

$$\begin{eqnarray}&&{M}_{\mathrm{metals}}={f}_{{\rm{Z}}}{\left\langle Z^{\prime} \right\rangle }_{V}{M}_{\mathrm{CGM}},\end{eqnarray} \tag{ 11 }$$

where ${f}_{{\rm{Z}}}=0.012$ is the mass fraction of metals at a solar metallicity, adopting the individual abundances from Asplund et al. (2009). The average line-of-sight metallicity is

$$\begin{eqnarray}&&\bar{Z^{\prime} }=\displaystyle \frac{1}{N}\int Z^{\prime} (r)n(r){ds},\end{eqnarray} \tag{ 12 }$$

where ds is the path element and N is the total gas column density along this sight line. The sight line can be calculated for an observer inside the galaxy (for MW observations) or an external observer at a given impact parameter (for other galaxies).

In solving Equations (1)–(7), we set r_b, the reference point for the boundary conditions of the gas distribution, at the outer radius of the corona ${r}_{\mathrm{CGM}}$. We now discuss the value ranges we consider for ${r}_{\mathrm{CGM}}$ and the gas properties there, such as the density and temperature, by estimating them for the MW.

Structure formation calculations and simulations predict that matter that falls onto the galaxy is shocked and heated (White 1978; Birnboim & Dekel 2003). We define the boundary between the IGM and CGM as the location of this accretion shock. Simulations indicate that this occurs roughly, but not exactly, at the virial radius (Schaal & Springel 2015), which is estimated from the halo total mass. The mass of the MW has been measured over the last decade using a variety of methods, resulting in ${M}_{\mathrm{vir}}=(1.3\pm 0.3)\times {10}^{12}$ ${M}_{\odot }$ (Bland-Hawthorn & Gerhard 2016). In FSM17, we used the gravitational potential profile from Klypin et al. (2002; model B; see their Table 2), which has ${r}_{\mathrm{vir}}=258\,\mathrm{kpc}$, and ${M}_{\mathrm{vir}}={10}^{12}$ ${M}_{\odot }$. These values are consistent with the range estimated by Bland-Hawthorn & Gerhard (2016), and we use the same gravitational potential and virial radius in this work.

We combine the uncertainties regarding the (i) size (and mass) of the MW halo (i.e., ${r}_{\mathrm{vir}}$) and (ii) location of the accretion shock into the range for ${r}_{\mathrm{CGM}}$ and examine values between the virial radius and 1.3 ${r}_{\mathrm{vir}}$, or $\sim 260\mbox{--}330\,\mathrm{kpc}$. Smaller CGM radii are not implausible in theory, but they may be inconsistent with measurements of O vi in other L* galaxies, as we discuss in Section 5.2 (see Johnson et al. 2015).

We now turn to the gas properties at this radius. First, we set the temperature, ${T}_{\mathrm{th}}({r}_{\mathrm{CGM}})$, to the virial temperature, defined by $2{E}_{{\rm{k}}}={E}_{\mathrm{pot}}$, where ${E}_{\mathrm{pot}}$ is the potential energy of the mean particle evaluated at the outer boundary. The gas temperature is then given by

$$\begin{eqnarray}&&{\text{}}{T}_{\mathrm{vir}}=\displaystyle \frac{G\bar{m}}{3{k}_{{\rm{B}}}}\displaystyle \frac{M(r)}{r}.\end{eqnarray} \tag{ 13 }$$

Scaling this to the Galaxy mass and ${r}_{\mathrm{CGM}}$, we get

$$\begin{eqnarray}&&{\text{}}{T}_{\mathrm{vir}}\approx 3.4\times {10}^{5}\,{\rm{K}}\left(\displaystyle \frac{\bar{m}}{0.59\,{m}_{{\rm{p}}}}\right)\left(\displaystyle \frac{{M}_{\mathrm{vir}}}{{10}^{12}\,{M}_{\odot }}\right){\left(\displaystyle \frac{{r}_{\mathrm{CGM}}}{300\mathrm{kpc}}\right)}^{-1},\end{eqnarray} \tag{ 14 }$$

where $0.59{m}_{{\rm{p}}}$ is the mean particle mass for fully ionized gas with the primordial abundance of helium. Birnboim & Dekel (2003) performed a detailed calculation of the gas temperature behind the virial shock and found a similar value. In this work, we consider temperatures in the range ${T}_{\mathrm{th}}({r}_{\mathrm{CGM}})\approx (2\mbox{--}4)\times {10}^{5}$ K, accounting for the uncertainty in the MW mass and the location of the shock. At these temperatures, the O vi ion fraction, ${f}_{{\rm{O}}{\rm{VI}}}$, is close to its peak in collisional ionization equilibrium (CIE), with ${f}_{{\rm{O}}{\rm{VI}}}\approx 0.25$ at ${T}_{\mathrm{peak}}\approx 3\times {10}^{5}$ K (Gnat & Sternberg 2007 and Section 3.2 here).

In our new model, each of the components providing pressure support behaves differently with radius due to a different EoS or adiabatic index, and the α parameter is a function of radius. For α at ${r}_{\mathrm{CGM}}$, we consider a range between 1 and 3, as we did in FSM17. For $\alpha =1$, there is only thermal and turbulent support, while pressure equipartition between thermal, magnetic, and cosmic rays gives $\alpha =3$ (see also Kempski & Quataert 2020).

For the turbulent velocity scale, we adopt ${\sigma }_{\mathrm{turb}}\sim 60$ $\mathrm{km}\ {{\rm{s}}}^{-1}$, similar to FSM17 (see Section 2.1 and Table 3 there). This velocity was estimated from the velocity dispersion of the O vi absorption features in the COS (Cosmic Origins Spectrograph on the Hubble Space Telecope) star-forming (SF) galaxies, reported by Tumlinson et al. (2011a). In our model, the O vi traces the extended warm/hot CGM.

To estimate the gas density at ${r}_{\mathrm{CGM}}$, we consider the conditions inside and outside the MW halo. McConnachie et al. (2007) inferred a lower limit of ${10}^{-5}\mbox{--}{10}^{-6}$ cm⁻³ for the Local Group (LG) intragroup medium density from ram pressure stripping of the Pegasus dwarf galaxy at $d\approx 920\,\mathrm{kpc}$ from the MW. Faerman et al. (2013) used the H i distribution in Leo T to estimate an upper limit for the gas pressure in the LG. They found that at d = 420 kpc from the MW, ${P}_{\mathrm{IGM}}/{k}_{{\rm{B}}}\,\lesssim 150$ K cm⁻³. Assuming that the pressure of the intragroup medium in the LG does not vary significantly with position on a 100 kpc scale gives an estimate for the CGM density:

$$\begin{eqnarray}&&n({r}_{\mathrm{CGM}})\sim \displaystyle \frac{{P}_{\mathrm{IGM}}}{\alpha ({r}_{\mathrm{CGM}}){T}_{\mathrm{th}}({r}_{\mathrm{CGM}})+{\sigma }_{\mathrm{turb}}^{2}\bar{m}/{k}_{{\rm{B}}}}.\end{eqnarray} \tag{ 15 }$$

For the chosen ${T}_{\mathrm{th}}({r}_{\mathrm{CGM}})$, the adopted range of $\alpha ({r}_{\mathrm{CGM}})$ and ${\sigma }_{\mathrm{turb}}$, this gives an upper limit of ${n}_{{\rm{H}}}({r}_{\mathrm{CGM}})\lt (0.5\mbox{--}2)\times {10}^{-4}$ cm⁻³, where ${n}_{{\rm{H}}}$ is the hydrogen volume density. Another estimate is obtained at smaller distances from the Galaxy. As discussed in FSM17 (see Section 5.1 there), studies of ram pressure stripping in the Large Magellanic Cloud (LMC) and MW dwarf satellite galaxies find CGM densities of $\sim {10}^{-4}$ cm⁻³ at $50\mbox{--}100\,\mathrm{kpc}$ (Grcevich & Putman 2009; Gatto et al. 2013; Salem et al. 2015), and Blitz & Robishaw (2000, hereafter BR00) found an average density of $\sim 2.4\times {10}^{-5}$ cm⁻³ inside 250 kpc. These values serve as upper limits for the density at the outer boundary, and we consider densities of ${n}_{{\rm{H}}}({r}_{\mathrm{CGM}})\sim (1\mbox{--}5)\times {10}^{-5}$ cm⁻³.

For the metallicity, we examine values in the range of ${Z}_{0}^{{\prime} }=0.5\mbox{--}1$ at the solar radius and $0.1\mbox{--}0.5$ at ${r}_{\mathrm{CGM}}$. We set the upper limit at ${r}_{\mathrm{CGM}}$ as $Z^{\prime} =0.5$ to allow for a constant metallicity profile, for comparison with FSM17. The length scale increases from ${r}_{Z}\sim 30\,\mathrm{kpc}$ for a large metallicity gradient, ranging between Z = 0.1 and 1, to ${r}_{Z}\gt 250\,\mathrm{kpc}$ for flat metallicity profiles, changing by $\lesssim 25 \%$. For a metallicity profile that varies by a factor of 3–5 between small radii and ${r}_{\mathrm{CGM}}$, the length scale is ${r}_{Z}\approx 50\mbox{--}100\,\mathrm{kpc}$. These scales are similar to the extent of galactic winds in numerical simulations (Salem et al. 2015; Fielding et al. 2017), and we prefer them in our model.

To summarize, the combination of ${n}_{{\rm{H}}}({r}_{\mathrm{CGM}})$, ${T}_{\mathrm{th}}({r}_{\mathrm{CGM}})$, $\alpha ({r}_{\mathrm{CGM}})$, and ${\sigma }_{\mathrm{turb}}$ allows us to compute the entropy parameters (Equations (6)–(7)), numerically solve Equation (4), and obtain the gas density profile, $\rho (r)$. Then, using the EoS, we get the individual and total pressure and temperature profiles from the outer boundary to the inner radius at the solar circle at ${R}_{0}=8.5\,\mathrm{kpc}$. This radius is the inner boundary in our model. In FSM17, we estimated that the thermal pressure above the Galactic disk, ${P}_{\mathrm{th}}({R}_{0})$, is between ∼1000 and 3000 K cm⁻³ (Wolfire et al. 2003; Dedes & Kalberla 2010). Putman et al. (2012) found pressures of $P/{k}_{{\rm{B}}}\sim 500\mbox{--}1300$ K cm⁻³ using observations of high-velocity clouds (HVCs) at distances of $10\mbox{--}15\,\mathrm{kpc}$ from the Galactic center (GC) and $3\mbox{--}9\,\mathrm{kpc}$ above the disk. With the above observational constraints in mind, we set the boundary conditions by fixing the temperature at the outer radius (${r}_{\mathrm{CGM}}$) and varying the density and nonthermal support there to set the inner pressure at ${R}_{0}$. Then, the metallicities at ${R}_{0}$ and ${r}_{\mathrm{CGM}}$ determine the metallicity length scale, and the distribution of metals is given by Equation (8).

Figure 1 presents the total hydrogen density and the thermal temperature profiles in the corona (left and right panels, respectively).⁷ For these models, we adopt ${r}_{\mathrm{CGM}}=1.1{r}_{\mathrm{vir}}\,=283\,\mathrm{kpc}$. In our fiducial model, the density and temperature at this boundary are ${n}_{{\rm{H}}}({r}_{\mathrm{CGM}})=1.1\times {10}^{-5}$ cm⁻³ and ${T}_{\mathrm{th}}({r}_{\mathrm{CGM}})=2.4\times {10}^{5}$ K. Both increase inward and at ${R}_{0}$ are equal to $2.9\times {10}^{-4}$ cm⁻³ and $2.1\times {10}^{6}$ K, respectively. The mean hydrogen density within ${r}_{\mathrm{CGM}}$ is $1.8\times {10}^{-5}$ cm⁻³.

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The density, temperature, and pressure profiles are well approximated by power-law functions of the radius, $\tilde{p}\times {\left(r/{r}_{\mathrm{CGM}}\right)}^{-a}$, where $\tilde{p}$ is the value of the fit at ${r}_{\mathrm{CGM}}$. Fits between ${R}_{0}$ and ${r}_{\mathrm{CGM}}$ give indices of ${a}_{{\rm{n}}}=0.93$ and ${a}_{{\rm{T}}}=0.62$ for the density and temperature, respectively. These approximations are accurate to within 20% for the density and 10% for the temperature, and they are shown in Figure 1 as dotted curves. The full approximations, including the normalization factors, are given in Table 1.

Figure 2 shows the pressure versus radius. The left panel shows the total and thermal pressures. The total pressure (black curve) at ${r}_{\mathrm{CGM}}$ is $P/{k}_{{\rm{B}}}=20$ K cm⁻³. This value is consistent with pressure estimates from the accretion rate onto an MW-like galaxy in cosmological simulations. The thermal pressure (red curve) at ${R}_{0}$ is 1350 K cm⁻³. This is near the lower limit of the range estimated in FSM17 from observations to be between ∼1000 and 3000 K cm⁻³ (see Section 2 there). The power-law index of the (total) pressure profile is ${a}_{{\rm{P}}}=1.35$, and this approximation is accurate to within 10%.

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The right panel of Figure 2 shows the fractional contribution of each pressure component as a function of radius: thermal support (red), nonthermal pressure from magnetic fields and cosmic rays (blue), and turbulent pressure (green). The ratio of nonthermal to thermal pressure is parameterized by ${P}_{\mathrm{nth}}/{P}_{\mathrm{th}}=\alpha -1$, and to include the contribution of turbulent support, we define ${\alpha }_{\mathrm{OML}}={P}_{\mathrm{tot}}/{P}_{\mathrm{th}}$. With $\alpha ({r}_{\mathrm{CGM}})=2.1$ and ${\alpha }_{\mathrm{OML}}({r}_{\mathrm{CGM}})=3.2$, the three components have similar contributions to the total pressure at ${r}_{\mathrm{CGM}}$. However, due to the higher adiabatic index of the thermal component, the thermal pressure increases more rapidly at smaller radii and dominates the total pressure at $r\lt 50\,\mathrm{kpc}$, with $\alpha ({R}_{0})\sim 1.5$.

We can estimate the strength of the magnetic field implied by the nonthermal pressure. If the cosmic rays and magnetic fields have equal contributions to the energy density, the magnetic field strength in the CGM is given by $B=\sqrt{4\pi {P}_{\mathrm{nth}}}\propto {r}^{-2{a}_{n}/3}$. In our fiducial model, this results in $B\propto {r}^{-0.62}$, and the field increases from $B\approx 110$ nG at ${r}_{\mathrm{CGM}}$ to 940 nG at ${R}_{0}$. These values are consistent with the upper limit inferred by Prochaska et al. (2019) for the magnetic field in the CGM of a massive galaxy at $z\approx 0.36$, with $B\lesssim 500$ nG at an impact parameter of $\approx 30\,\mathrm{kpc}$.

Given the density profile, we calculate the CGM mass and its contribution to the baryonic budget of the Galaxy. The cumulative gas mass distribution is shown in Figure 4 for spherically enclosed and projected masses (red solid and dashed curves, respectively). The coronal gas mass inside ${r}_{\mathrm{vir}}$ is $4.6\times {10}^{10}$ ${M}_{\odot }$. Adopting a cosmological baryon fraction of ${f}_{\mathrm{bar}}=0.157$ (Planck Collaboration et al. 2016), this constitutes $\sim 30 \%$ of the Galactic baryonic budget. Together with a mass of $\approx 6\times {10}^{10}$ ${M}_{\odot }$ for the Galactic disk (McMillan 2011; Licquia & Newman 2015), we get a total baryonic mass of $\sim 1.1\times {10}^{11}$ ${M}_{\odot }$, or $\sim 70 \%$ of the Galactic baryons expected within ${r}_{\mathrm{vir}}$. The total CGM mass inside ${r}_{\mathrm{CGM}}$ is $5.5\times {10}^{11}$ ${M}_{\odot }$.

The gas phase metallicity in our model decreases from $Z^{\prime} =1.0$ at ${R}_{0}$ to 0.3 at ${r}_{\mathrm{CGM}}$ with a metallicity scale length of ${r}_{Z}=90\,\mathrm{kpc}$ and is plotted in Figure 3. The total mass of metals within ${r}_{\mathrm{CGM}}$ is $3.1\times {10}^{8}$ ${M}_{\odot }$. The cumulative metal mass profiles are shown in Figure 4 for the spherically enclosed and projected masses (black solid and dashed curves, respectively). These can be useful for comparison with mass estimates from measurements of metal ion column densities (see Section 5).

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For example, Peeples et al. (2014, hereafter P14) analyzed the COS-Halos O vi measurements to estimate the metal gas mass in the warm CGM. For their preferred model, with an assumed density profile slope of ${a}_{{\rm{n}}}=2$, they inferred the projected metal mass inside 150 kpc and obtained ${M}_{\mathrm{metals}}(h\lt 150\,\mathrm{kpc})\,\sim 0.46\times {10}^{8}$ ${M}_{\odot }$, with a range of $(0.28\mbox{--}1.1)\times {10}^{8}$ ${M}_{\odot }$. However, P14 found that ${a}_{{\rm{n}}}$ has a significant effect on the total gas mass. For a profile with a slope of ${a}_{{\rm{n}}}=1$, a higher-mass profile is allowed by the measurements, with ${M}_{\mathrm{metals}}(h\,\lt 150\,\mathrm{kpc})\sim 4\times {10}^{8}$ ${M}_{\odot }$. In our fiducial model, the projected metal mass in the gas phase within 150 kpc is $1.9\times {10}^{8}$ ${M}_{\odot }$, within the range allowed by the different gas density distributions in P14.

Peek et al. (2015) found that galactic coronae contain significant amounts of dust and estimated a dust mass of ${M}_{\mathrm{dust}}\sim 6\times {10}^{7}$ ${M}_{\odot }$ in the CGM of $0.1\mbox{--}1.0\,{L}^{* }$ galaxies. This is lower than the gas phase metal mass in our model, but not negligible. Our model does not constrain the dust content of the CGM.

The warm/hot gas at each radius in our model is at a constant temperature and density given by the profiles presented in Figure 1. In computing the ionization fractions, we include electron-impact collisional ionization and photoionization by the MGRF. We assume ionization equilibrium. We do not include photoionization by stellar radiation from the Galaxy, since stellar radiation is expected to decrease rapidly as ${d}^{-2}$ with the distance d from the Galaxy and is not energetic enough to affect the high oxygen ions we address here (O vi–O viii). Other Galactic sources may have a contribution to ionizing radiation, although probably on lower ions and at small distances (see Cantalupo 2010 and Upton Sanderbeck et al. 2018). The MW and the COS-Halos galaxies do not have AGNs, and we do not include AGN radiation fields (although see Oppenheimer et al. 2018 for consideration of "fossil" O vi AGN photoionization).

In CIE, the ion fractions are functions of the gas temperature only (Gnat & Sternberg 2007). When photoionization is included, the ion fractions may also depend on the gas density and radiation field properties, such as intensity and spectral shape (Gnat 2017). For a field with a known spectral distribution, the effect of the radiation on the atomic ionization state can be estimated using the ionization parameter, given by $U\equiv {\rm{\Phi }}/{{cn}}_{{\rm{H}}}$. Here ${\rm{\Phi }}=4\pi {\int }_{{\nu }_{0}}^{\infty }\tfrac{{J}_{\nu }}{h\nu }d\nu$ is the ionizing photon flux, ${J}_{\nu }$ is the radiation field energy flux density, and c is the speed of light. In our calculations, we consider the Haardt & Madau (2012, hereafter HM12) radiation field, which is a function of redshift only. For the HM12 z = 0 MGRF, ${\rm{\Phi }}\approx {10}^{4}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. Scaling the ionization parameter to this value and the gas density at the outer boundary of our corona model (see Table 1), we get

$$\begin{eqnarray}&&U=3.3\times {10}^{-2}\left(\displaystyle \frac{{\rm{\Phi }}}{{10}^{4}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{-5}{\mathrm{cm}}^{-3}}\right)}^{-1}.\end{eqnarray} \tag{ 16 }$$

At z = 0.2, the median redshift of the COS-Halos galaxies, the ionizing photon flux of the HM12 field is ${\rm{\Phi }}\approx 2.3\,\times {10}^{4}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, and we continue our calculation for z = 0.2.

The MGRF intensity in the EUV and soft X-rays is uncertain to some extent, with different studies arguing for a stronger (Stern et al. 2018; Faucher-Giguère 2020) or weaker (Bland-Hawthorn et al. 2017) radiation field. In this work, we adopt the HM12 spectrum and note that the FG20 and HM12 field intensities are within $\sim 30 \%$ of each other between 0.1 and 2 keV, the energy range relevant for the ions we address here. The differences may be larger at lower energies and are more relevant for lower ions (see Werk et al. 2016; Prochaska et al. 2017; Upton Sanderbeck et al. 2018).

The gray contours in Figure 5 show the O vi ion fraction, ${f}_{{\rm{O}}{\rm{VI}}}$, in the temperature–density parameter space, calculated in the presence of the z = 0.2 HM12 MGRF using Cloudy 17.00 (Ferland et al. 2017). At hydrogen densities above ${n}_{{\rm{H}}}\sim {10}^{-3}$ cm⁻³, the ion fraction is set by collisional ionization. It is then a function of the gas temperature only and peaks at ${T}_{\mathrm{peak},{\rm{O}}{\rm{VI}}}\sim 3\times {10}^{5}$ K, with ${f}_{{\rm{O}}{\rm{VI}}}\approx 0.25$. The O vi ion fraction at temperatures far from this peak, at $T\lt {10}^{5}$ K ($T\gt {10}^{6}$ K), is low, and oxygen exists in lower (higher) ionization states. At lower densities, below ${n}_{{\rm{H}}}\sim {10}^{-5}$ cm⁻³, the ion fractions clearly deviate from their CIE values due to photoionization.

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In general, radiation increases the overall gas ionization, but the change in the fraction of a specific ion, ${f}_{\mathrm{ion}}$, depends on the gas temperature compared to ${T}_{\mathrm{peak},\mathrm{ion}}$. For $T\lt {T}_{\mathrm{peak},\mathrm{ion}}$, energetic photons ionize the lower ionization states and increase ${f}_{\mathrm{ion}}$ compared to CIE. In gas at higher temperatures, radiation ionizes the atom to a higher state and reduces ${f}_{\mathrm{ion}}$.

We define ${U}_{\mathrm{crit}}$ as the threshold ionization parameter above which an ion fraction deviates by more than 10% compared to the CIE value. While the threshold can vary with temperature, for our qualitative analysis here for the O vi, we adopt a single value of ${U}_{\mathrm{crit},{\rm{O}}{\rm{VI}}}\sim 7\times {10}^{-3}$. At z = 0.2, this corresponds to a density of ${n}_{{\rm{H}},\mathrm{photo}}\sim {10}^{-4}$ cm⁻³, below which photoionization is important. In Figure 5, this critical density is indicated by the vertical green dashed line.

The red line shows the $T\propto {n}^{2/3}$ temperature–density relation in our model (with $\gamma =5/3$ for thermal pressure). The black squares mark specific radii between ${r}_{\mathrm{CGM}}$ and the solar circle. For $r\gtrsim 30\,\mathrm{kpc}$, our model has densities close to the critical photoionization density of 10⁻⁴ cm⁻³. To compare the ion fraction at a given radius in the model to the fraction in CIE, one can move horizontally (at $T=\mathrm{const}.$) from the red curve to a density 1–2 dex above the photoionization threshold and estimate how the ion fraction changes along this line. Since most of the gas in our model is above ${T}_{\mathrm{peak},{\rm{O}}{\rm{VI}}}=3\times {10}^{5}$ K, photoionization reduces the O vi fraction compared to CIE. This is in contrast with models at lower temperatures, where photoionization is invoked as an O vi production mechanism (e.g., Stern et al. 2018).

The O vii and O viii ions have photoionization densities similar to the O vi (see also Ntormousi & Sommer-Larsen 2010), and in our model, they are also affected by the MGRF. The O vi–O viii ion fractions as functions of radius in our model are plotted in the left panel of Figure 6. We also display the N v fraction and discuss these curves in more detail in Section 3.3.

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The total ion densities, ${n}_{\mathrm{ion}}={f}_{\mathrm{ion}}{A}_{{\rm{i}}}Z^{\prime} {n}_{{\rm{H}}}$, are also a function of the gas density and metallicity profiles and the elemental abundances ${A}_{{\rm{i}}}$. The volume densities of O vi–O viii and N v are shown in the right panel of Figure 6 as a function of radius. In Section 5 we discuss the behavior of the column densities of these ions and compare them to observations.

We note that the measured O vii and O viii column densities are associated with the MW at z = 0. Comparing them to the results of our model, for which we adopt the MGRF at z = 0.2, may seem inconsistent. However, these column densities, observed from inside the Galaxy, form mostly in the inner, denser part of the corona ($r\lt 30$ kpc), where their ion fractions are set by the gas temperature only (see Figure 6). Thus, using the z = 0 MGRF has a very small effect on the O vii and O viii columns, and our comparison is valid (see also Section 5.1.1).

We conclude that for the gas properties of our fiducial model, photoionization by the MGRF has a nonnegligible effect on the ion fractions of the high oxygen ions (O vi–O viii). We calculate the metal ion fractions as a function of the gas density and temperature using Cloudy 17.00 (Ferland et al. 2017) and use them to calculate the ion volume and column densities, which we discuss in Sections 3.3 and 5. The MGRF also affects the gas radiative properties through the metal ion fractions, and we calculate the gas net cooling rate and emission spectrum as a function of the density, temperature, and metallicity. In Section 3.4, we use these quantities to calculate the emission properties of the corona.

Observations of the CGM reveal the gas distribution through absorption and emission by metal ions, and in this section, we describe the spatial distributions shown in Figure 6. We plot the O vi (solid blue), O vii (solid green), O viii (solid red), and N v (solid cyan) ions. The ion fractions are shown in the left panel with the CIE fractions (dashed curves) for comparison, and the ion volume densities are shown on the right. For the nitrogen and oxygen ion densities, we use the Asplund et al. (2009) abundances, with ${A}_{{\rm{N}}}=6.8\times {10}^{-5}$ and ${A}_{{\rm{O}}}=4.9\times {10}^{-4}$ for solar metallicity.

The N v and O vi ion fractions are most abundant at large radii ($r\gtrsim 150$ kpc), where the gas temperature, with $T\sim 3\times {10}^{5}$ K, is closest to their CIE peak temperatures ($\approx 2$ and $3\times {10}^{5}$ K, respectively). The gas density of the CGM at these radii is $\sim 3\times {10}^{-5}$ cm⁻³, so photoionization is significant and reduces the fractions of both ions. The O vi peak ion fraction, with ${f}_{{\rm{O}}{\rm{VI}}}\sim 0.1$, is close to its maximum at CIE ($f\approx 0.25$). The gas temperature is above ${T}_{\mathrm{peak},{\rm{N}}{\rm{V}}}$, and this, together with photoionization, leads to lower ion fractions compared to O vi, with ${f}_{{\rm{N}}{\rm{V}}}\lesssim 0.02$.

The effect of photoionization on O vii varies with radius. At intermediate radii, $30\mbox{--}150\,\mathrm{kpc}$, the gas temperatures are such that O vii is abundant in CIE, with ${f}_{{\rm{O}}{\rm{VII}}}\sim 1$, and photoionization reduces the O vii fraction, but the effect is small ($10 \% \mbox{--}20 \%$). At larger radii, where the temperature is below $\sim 5\times {10}^{5}$ K, photoionization increases the O vii fraction compared to its CIE values. The O viii is affected more significantly. In our fiducial model, the gas temperature is high enough to form O viii collisionally in the central part ($r\lesssim 25$ kpc). Photoionization by the MGRF creates O viii at larger radii, and its ion fraction increases with the ionization parameter, from ${f}_{{\rm{O}}{\rm{VIII}}}\sim 0.1$ at 30 kpc to ∼0.2 at ${r}_{\mathrm{CGM}}$. Overall, O vii is the dominant oxygen ion in our model at all radii and almost equal to the O viii fraction at the solar circle.

The resulting densities for our four ions of interest are plotted in the right panel of Figure 6. The O vii and O viii ion fractions do not vary strongly with radius. This leads to decreasing ion volume densities as a result of the density and metallicity distributions in the model. The N v and O vi fractions, on the other hand, increase with radius, resulting in more extended distributions with almost flat ion density profiles. The O vi volume density is in the range ${n}_{{\rm{O}}{\rm{VI}}}\sim (2\mbox{--}4)\times {10}^{-10}$ cm⁻³ for radii between 10 and 250 kpc. The nitrogen abundance is ∼7 times lower than that of oxygen, and, together with the N v lower ion fraction, this gives volume densities of ${n}_{{\rm{N}}{\rm{V}}}\sim (4\mbox{--}8)\times {10}^{-12}$ cm⁻³, 20–60 times lower than those of O vi. We use the volume densities to calculate the ion column densities through the CGM, and in Section 5 we compare these to the measured values and limits.

The left panel of Figure 7 shows the predicted emission (cooling) spectrum of the CGM in our fiducial model. The spectrum is given by

$$\begin{eqnarray}&&J(\nu )=4\pi {\int }_{{R}_{0}}^{{r}_{\mathrm{CGM}}}{j}_{\nu }(n,T,Z^{\prime} ){r}^{2}{dr},\end{eqnarray} \tag{ 17 }$$

where j_ν is the emissivity ($\mathrm{erg}\,{\mathrm{cm}}^{-3}\ \,{\mathrm{Hz}}^{-1}\,{\mathrm{sr}}^{-1}$) of each gas parcel. We used Cloudy 17.00 (Ferland et al. 2017) to calculate the (optically thin) emissivities as functions of n, T, and $Z^{\prime}$ for gas subjected to photoionization by the z = 0.2 HM12 MGRF. (In FSM17, we assumed pure CIE for the emissivities.) The resulting spectrum in Figure 7 (denoted by L_ν) is displayed in units of $\mathrm{erg}\ {{\rm{s}}}^{-1}\,{\mathrm{keV}}^{-1}\,{\mathrm{sr}}^{-1}$ and consists of collisionally excited metal ion emission lines, recombination radiation, and bremsstrahlung. The red and black lines show the full and smoothed spectra, respectively. As shown in Figure 4, the gas in our model is dominated by large radii, where the gas temperature is low, and most of the emission is in the UV. The vertical dashed lines show the $0.4\mbox{--}2.0\,\mathrm{keV}$ band.

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The total luminosity⁸ of the warm/hot gas is given by

$$\begin{eqnarray}&&{L}_{{\rm{r}}{\rm{a}}{\rm{d}}}=\int J(\nu )d\nu \equiv 4\pi \int {\mathscr{L}}{r}^{2}dr,\end{eqnarray} \tag{ 18 }$$

where ${\mathscr{L}}$ is the radiative cooling rate per unit volume ($\mathrm{erg}\ {{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-3}\$) for each parcel. For our fiducial model, ${L}_{\mathrm{rad}}=9.4\times {10}^{40}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$. The emission in the $0.4\mbox{--}2.0\,\mathrm{keV}$ band is $\sim {10}^{39}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$, only $\sim 2 \%$ of the total cooling luminosity. We also compute the local net cooling rates given by ${\mathscr{L}}-{\mathscr{H}}={n}_{e}{n}_{{\rm{H}}}{\rm{\Lambda }}$, where ${\mathscr{H}}$ is the heating rate per unit volume⁹ due to the MGRF, and ${\rm{\Lambda }}\,(\mathrm{erg}\ {{\rm{s}}}^{-1}\,{\mathrm{cm}}^{3})$ is the net cooling efficiency. In CIE (${\mathscr{H}}=0$), Λ is a function of the gas temperature and metallicity only (Gnat & Sternberg 2007). In the presence of radiation, Λ is also a function of the ionization parameter and radiation spectral shape (Gnat 2017), and we use Cloudy to calculate it. We calculate the volume-integrated net cooling rate,

$$\begin{eqnarray}&&{L}_{{\rm{c}}{\rm{o}}{\rm{o}}{\rm{l}}}=4\pi \int {n}_{e}{n}_{{\rm{H}}}{\rm{\Lambda }}{r}^{2}dr,\end{eqnarray} \tag{ 19 }$$

and find that in our fiducial model, ${L}_{\mathrm{cool}}=7.6\times {10}^{40}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$. This implies that for our model, 20% of the emitted luminosity is reprocessed MGRF energy.

We integrate the spectrum in different energy bands along lines of sight through the corona to obtain the projected luminosity as a function of the impact parameter, and the result is shown in the right panel of Figure 7. The total projected emission profile (black curve) is extended, with a half-flux radius of ${r}_{1/2}\sim 100\,\mathrm{kpc}$. The $0.4\mbox{--}2.0\,\mathrm{keV}$ emission (solid magenta curve) comes from the hotter gas at smaller radii and is more centrally concentrated, with ${r}_{1/2}\sim 59\,\mathrm{kpc}$ (marked by the vertical dotted line). We note that the instrumental sensitivity and background emission in the X-ray is at the level of the predicted emission. Li et al. (2018, hereafter L18) performed a stacking analysis of the X-ray emission from massive galaxies in the local universe and estimated a background level of $I\sim {10}^{35}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{kpc}}^{-2}$. This threshold is shown in our plot by a horizontal dashed magenta line. We calculate the half-flux radius of the emission above this threshold and find a value of ${r}_{1/2}\sim 9\,\mathrm{kpc}$, marked by the vertical dashed line in the plot. This demonstrates the challenge in detecting the CGM of MW-like galaxies in emission, given the current instrumental sensitivity and background emission. The other solid curves in the plot show the projected emission in different energy bands: $E\lt 13.6\,\mathrm{eV}$ (blue), $13.6\mbox{--}400\,\mathrm{eV}$ (cyan), and $2\mbox{--}10\,\mathrm{keV}$ (red). As mentioned above, the total emission is dominated by the UV.

We now show that the detection of O vi in warm/hot gas implies an upper limit on the gas cooling time. We present a brief version of the analysis here and defer the full derivation to the Appendix. This result is not limited to our model and is relevant for a range of gas distributions.

To obtain an empirical upper limit, we relate the gas cooling time to the O vi column density. The latter is given by

$$\begin{eqnarray}&&{N}_{{\rm{O}}{\rm{VI}}}(h)=2{A}_{{\rm{O}}}{\int }_{0}^{z}{n}_{{\rm{H}}}(r)Z^{\prime} (r){f}_{{\rm{O}}{\rm{VI}}}(r){dz}^{\prime} ,\end{eqnarray} \tag{ 21 }$$

where h is the impact parameter and $z^{\prime} =\sqrt{{r}^{2}-{h}^{2}}$.¹¹ Assuming the gas properties vary as power-law functions of the radius, we can rewrite this as

$$\begin{eqnarray}&&{N}_{{\rm{O}}{\rm{VI}}}(h)=2{A}_{{\rm{O}}}{f}_{{\rm{O}}{\rm{VI}}}(h){n}_{{\rm{H}}}(h)Z^{\prime} (h){{RI}}_{a},\end{eqnarray} \tag{ 22 }$$

where R is the outer radius of the gas distribution (i.e., ${r}_{\mathrm{CGM}}$ in our model) and I_a is a dimensionless integral of order unity for a range of power-law slopes for ${f}_{{\rm{O}}{\rm{VI}}}$, ${n}_{{\rm{H}}}$, and $Z^{\prime}$.

The key step is to isolate the product, ${n}_{{\rm{H}}}Z^{\prime}$, in Equation (22) and insert into the cooling time. This gives

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}(h)=5.8{A}_{{\rm{O}}}\left[\displaystyle \frac{{k}_{{\rm{B}}}T(h){f}_{{\rm{O}}{\rm{VI}}}(h)}{{{\rm{\Lambda }}}_{\odot }(T,n)}\right]\displaystyle \frac{{{RI}}_{a}}{{N}_{{\rm{O}}{\rm{VI}}}(h)},\end{eqnarray} \tag{ 23 }$$

where we used ${\rm{\Lambda }}={{\rm{\Lambda }}}_{\odot }(T,n)Z^{\prime}$, ignoring cooling by hydrogen and helium and resulting in a lower limit for the cooling time. Given the shape of the cooling function and the O vi ion fraction in the temperature–density space in the presence of the HM12 MRGF at z = 0.2, the term in square brackets is bound from above for gas at $T\gt {10}^{5}$ K, with ${k}_{{\rm{B}}}{Tf}_{{\rm{O}}{\rm{V}}{\rm{I}}}/{{\rm{\Lambda }}}_{\odot }\leqslant 4.6\times {10}^{10}\,{\rm{s}}\,{{\rm{c}}{\rm{m}}}^{-3}\,$. The maximum occurs at $T\sim 3.5\times {10}^{5}$ K and densities above ${n}_{{\rm{H}}}\geqslant {10}^{-4}$ cm⁻³, where ${f}_{{\rm{O}}{\rm{VI}}}$ is maximal (see Section 3.2). At lower densities, radiation suppresses the cooling function, but the O vi fraction is reduced even more, so that overall, the term is smaller.¹² Finally, ${I}_{a}=0.5\pm 0.18$ dex for power-law slopes between 0.5 and 2.5 and impact parameters in the range $0.3\lt h/R\lt 0.9$ (see the Appendix). Inserting these into Equation (23), we get

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}(r=h)\lesssim 5.6\,\left(\displaystyle \frac{R}{260\,\mathrm{kpc}}\right){\left(\displaystyle \frac{{N}_{{\rm{O}}{\rm{VI}}}(h)}{3\times {10}^{14}{\mathrm{cm}}^{-2}}\right)}^{-1}\,\mathrm{Gyr},\end{eqnarray} \tag{ 24 }$$

where we scaled the corona size to the median virial radius of the COS-Halos galaxies and the O vi column density to the typical O vi column density measured at $h/{r}_{\mathrm{vir}}\approx 0.6$ by Tumlinson et al. (2011a; see Figure 10). The cooling time range resulting from variation in the underlying gas distributions affecting the value of I_a is ±30%.

For the halo dynamical time in Equation (20), we fit the mass distribution in the Klypin profile at large radii (where it is dominated by a Navarro–Frenk–White (NFW) profile) as $M\propto {r}^{0.56}$ and scale it to the MW, resulting in

$$\begin{eqnarray}&&{t}_{\mathrm{dyn}}(r)\approx 2.8\,{\left(\displaystyle \frac{r}{260\mathrm{kpc}}\right)}^{1.22}\,\mathrm{Gyr}.\end{eqnarray} \tag{ 25 }$$

We define the ratio $\zeta \equiv {t}_{\mathrm{cool}}/{t}_{\mathrm{dyn}}$ and combine Equations (24) and (25) to give

$$\begin{eqnarray}&&\zeta (r=h)\lt 2.0\,{\left(\displaystyle \frac{h}{260\mathrm{kpc}}\right)}^{-1.22}{\left(\displaystyle \frac{{N}_{{\rm{O}}{\rm{VI}}}(h)}{3\times {10}^{14}{\mathrm{cm}}^{-2}}\right)}^{-1}.\end{eqnarray} \tag{ 26 }$$

Our approximation and the derived upper limit are valid for $0.3\lt h/R\lt 0.9$, and the O vi column density we use is measured at 0.6 ${r}_{\mathrm{vir}}$. This implies $\zeta \lesssim 3.7$ (2.8–4.8 uncertainty range), much below the value of ∼10 estimated by Sharma et al. (2012a, 2012b) and Voit et al. (2017) for galaxy clusters. A ratio of $\zeta \sim 10$ would require O vi columns lower by a factor of $\sim 2\mbox{--}3$ than observed in the CGM of L* galaxies by COS-Halos.

The upper limit we derive applies to the gas responsible for the dominant part of the observed O vi absorption. For a separate, hotter component that contributes a small fraction of the total measured O vi column, the upper limit as given by Equation (23) will be higher. For example, in our FSM17 model, the O vi column forms mainly in gas that has cooled out of much hotter O vii and O viii absorbing gas. For the cooled component, ${t}_{\mathrm{cool}}\lt 1\,\mathrm{Gyr}$, and ${t}_{\mathrm{cool}}/{t}_{\mathrm{dyn}}\lt 1$, consistent with our upper limit. However, for the hot component, ${t}_{\mathrm{cool}}/{t}_{\mathrm{dyn}}\sim 6$ at large radii, slightly exceeding this limit. In the isentropic model in this paper, the bulk of the CGM gas mass is traced by O vi for which the upper limit is obtained.

If gas with $\zeta \lt 10$ is thermally unstable and develops multiphase structure, the upper limit we derive on the O vi-bearing gas cooling time implies that cool gas should be present when O vi is detected. The COS-Halos observations seem to be consistent with this prediction, with detections of H i and low metal ions (Werk et al. 2013, 2014; Prochaska et al. 2017). In Paper III (Y. Faerman et al. 2020, in preparation), we will extend our model to include a cool ($T\sim {10}^{4}$ K), purely photoionized gas component and compare it to existing observations.

Figure 8 shows the timescales in our fiducial model as a function of radius, calculated numerically using the definitions in Equation (20). The black curve is the dynamical time of the Galactic halo, given the Klypin et al. (2002) potential. The solid magenta curve is the gas cooling time. At large radii, $r\gt 100\,\mathrm{kpc}$, the two timescales are well approximated by power-law functions of the radius. The dynamical time is given by Equation (25), and the cooling time is

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}(r)\approx 6.5\,{\left(\displaystyle \frac{r}{{r}_{\mathrm{CGM}}}\right)}^{0.91}\,\,\mathrm{Gyr}.\end{eqnarray} \tag{ 27 }$$

This approximation is accurate to within 10% between 100 kpc and ${r}_{\mathrm{CGM}}$, and it is shown by the dotted magenta curve in Figure 8. The fit to the dynamical time is accurate to within 1% in the same range, and in the plot, the approximation is indistinguishable from the numerical calculation. Since the two timescales vary similarly with radius, their ratio is almost constant, increasing from $\zeta =2.4$ at ${r}_{\mathrm{CGM}}$ to 3.1 at 100 kpc. These values are consistent with the limit derived in Section 4.1 and significantly below the value of ∼10 estimated by Voit et al. (2017) in clusters of galaxies and adopted by Voit (2019) for the CGM (see also Stern et al. 2019).

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The mean global cooling timescale for the corona is the total thermal energy, ${E}_{\mathrm{th}}=8.6\times {10}^{57}$ erg, divided by the net cooling rate, ${L}_{\mathrm{cool}}=7.6\times {10}^{40}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$, giving $\left\langle {t}_{\mathrm{cool}}\right\rangle \equiv {E}_{\mathrm{th}}/{L}_{\mathrm{cool}}=3.6\,\mathrm{Gyr}$. As discussed in Section 2, our model assumes a steady state, so that (most of) the radiative losses are offset by heating (see Section 4.3 for energy budget estimates) and the CGM is stable on an ∼10 Gyr timescale. We now briefly discuss the gas cooling properties other than ${t}_{\mathrm{cool}}$, which may be useful to study the energy budget of the CGM.

Figure 9 shows the distribution of local and mass cooling rates from the CGM. The left panel shows the gas radiative cooling rate, ${\mathscr{L}}={n}_{{\rm{e}}}{n}_{{\rm{H}}}{\rm{\Lambda }}$, as a function of radius. The dashed curve shows the local rate per unit length, $4\pi {r}^{2}{\mathscr{L}}$, between 1.0 and $3.6\times {10}^{38}\,\mathrm{erg}\ {{\rm{s}}}^{-1}\,{\mathrm{kpc}}^{-1}$. This is the distribution of energy injection rate needed to keep the CGM in a steady state and is a constraint for the mechanisms that can provide this energy. The solid curve is the integrated cumulative value inside r, with a total of $7.6\times {10}^{40}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ inside ${r}_{\mathrm{CGM}}$.

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Given the gas mass distribution (Figure 4) and its cooling time (Figure 8), we can calculate the gas mass cooling rate as $\dot{M}(r)={M}_{\mathrm{gas}}(r)/{t}_{\mathrm{cool}}(r)$. Without energy input into the CGM, this gives an upper limit on the gas mass cooling out of the corona and accreting onto the galaxy (see also Joung et al. 2012; Armillotta et al. 2016; Stern et al. 2020). The right panel of Figure 9 shows the mass cooling rate, similar to the (energy) cooling rate in the left panel, with the local value per unit length as the dashed curve and the integrated value as the solid curve. The local rate is almost flat with radius, with a mass cooling rate of $d\dot{M}/{dr}\approx 0.05\,{M}_{\odot }\ {\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-1}$ between 70 kpc and ${r}_{\mathrm{CGM}}$. The global mass cooling rate, given by ${\dot{M}}_{\mathrm{tot}}={M}_{\mathrm{CGM}}/{L}_{\mathrm{cool}}$, is $\approx 13.3$ ${M}_{\odot }\ {\mathrm{yr}}^{-1}$. Integrating this over 10 Gyr gives $\approx 1.3\times {10}^{11}$ ${M}_{\odot }$, a factor of ∼2 higher than the z = 0 mass of the MW disk.

We now address the energy budget of the CGM in our model and the energy sources available for creating the warm/hot corona and maintaining it in a steady state.

First, we calculate the total energy in the CGM by integrating the different pressure components in our fiducial model (see Figure 2) over the volume of the corona:

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{CGM}} & = & \displaystyle \int \left({E}_{\mathrm{th}}+{E}_{\mathrm{nth}}+{E}_{\mathrm{turb}}\right){dV}\\ & = & \displaystyle \int \left(\displaystyle \frac{3}{2}{P}_{\mathrm{th}}+3{P}_{\mathrm{nth}}+\displaystyle \frac{3}{2}{\sigma }_{\mathrm{turb}}^{2}\bar{m}n\right){dV}.\end{array}\end{eqnarray} \tag{ 28 }$$

This gives a total of $2.9\times {10}^{58}$ erg, with 29% in thermal energy, 51% in CR/B, and 20% in turbulent energy. Given the gas radiative cooling rate (see Equation (19)), we can estimate how much energy was lost from the CGM. Taking ${t}_{\mathrm{life}}\sim 10\,\mathrm{Gyr}$ for the galaxy lifetime, we get ${E}_{\mathrm{cool}}={L}_{\mathrm{cool}}\times {t}_{\mathrm{life}}\approx 2.4\times {10}^{58}$ erg. Thus, a total of ${E}_{\mathrm{tot}}\equiv {E}_{\mathrm{CGM}}+{E}_{\mathrm{cool}}\approx 5.3\times {10}^{58}$ erg is required to form the CGM and balance its radiative losses over the lifetime of the galaxy.

Possible energy sources for the CGM include feedback from SNe and the central SMBH and accretion onto the halo from the IGM. All three mechanisms can (i) generate shocks that heat the gas and accelerate cosmic rays and (ii) drive turbulence that amplifies magnetic fields and heats the gas when it dissipates. The SNe- and SMBH-driven outflows can advect magnetic fields and cosmic rays from the galactic disk to the CGM. We use MW data to estimate the energy budget available to fuel the CGM.

We estimate the total energy injected into the CGM by SNe over the lifetime of the galaxy as

$$\begin{eqnarray}&&{E}_{\mathrm{SNe}}\approx {M}_{* ,\mathrm{MW}}\times {R}_{\mathrm{SN}}\times {E}_{\mathrm{SN},{\rm{s}}}\times {f}_{\mathrm{SN}},\end{eqnarray} \tag{ 29 }$$

where the terms on the right-hand side are the MW present-day stellar mass, the mean number of SNe per unit of stellar mass, the total energy for a single SN event, and the fraction of that energy that goes to the CGM. We take ${M}_{* ,\mathrm{MW}}=5\times {10}^{10}$ ${M}_{\odot }$, ${R}_{\mathrm{SN}}$ of one SN event per 100 ${M}_{\odot }$, and ${E}_{\mathrm{SN},{\rm{s}}}={10}^{51}$ erg. The coupling factor, ${f}_{\mathrm{SN}}$, is uncertain, and for ${f}_{\mathrm{CGM}}=0.1$, we get ${E}_{\mathrm{SNe}}\approx 5.0\times {10}^{58}$ erg. Next, we estimate the energy from SMBH feedback. The mass of the MW SMBH, Sgr A^*, is ${M}_{\mathrm{MW},\mathrm{SMBH}}=4.1\times {10}^{6}$ ${M}_{\odot }$ (Gravity Collaboration et al. 2019). Assuming ${f}_{\mathrm{SMBH}}=0.03$ of the rest mass was converted to kinetic energy by the accretion disk (Sadowski & Gaspari 2017), we get ${E}_{\mathrm{SMBH}}\approx 2.2\times {10}^{59}$ erg.

We estimate the contribution of gas accretion from the IGM onto the galactic halo to the energy budget. We do this by calculating the gravitational energy in the CGM, ${E}_{\mathrm{acc}}={f}_{\mathrm{acc}}\times \left({{GM}}_{\mathrm{halo}}/{r}_{\mathrm{CGM}}\right)\times {M}_{\mathrm{CGM}}$, where ${f}_{\mathrm{acc}}$ is a numerical coefficient of order unity accounting for the growth of the Galaxy with cosmic time. For our fiducial model, this results in ${E}_{\mathrm{acc}}/{f}_{\mathrm{acc}}\approx 1.7\times {10}^{58}$ erg. While it is subdominant compared to the energy outputs of SNe and the SMBH, accretion provides an energy source at the outer boundary of the halo that is different from SN and SMBH feedback, which injects energy at the base of the corona.

The energy from SNe and SMBHs is injected from within the galaxy. The observations show that the CGM at large radii ($\gt 100$ kpc) is metal-enriched (Tumlinson et al. 2011a; Prochaska et al. 2017), providing evidence for extended galactic outflows. Numerical simulations also suggest that winds driven by SN and SMBH feedback can transport energy and metals to large radii in the halo (McNamara & Nulsen 2007; Bower et al. 2017; Fielding et al. 2017; Li & Tonnesen 2019). However, we do not consider specific transport processes in this paper.

For the gas densities in our model, the equilibrium temperatures for heating by the MGRF are $\sim 2\times {10}^{4}$ K, significantly lower than the gas temperatures in the model. As noted in Section 3.4, only 20% of the total computed luminosity (shown in Figure 7) is reprocessed MGRF energy.

To summarize, our estimate shows that there is enough energy available in an MW-like galaxy to form and power the CGM. We have shown that SNe and energy injection by Sgr A* can offset the radiative losses over ∼10 Gyr. The energy from SN events is sufficient to power the CGM for a coupling constant of ${f}_{\mathrm{SN}}=0.1$. The energy from SMBH feedback is a factor of ∼4 higher than that from SNe, and if a significant fraction of this energy was injected into the CGM, it could have ejected gas beyond the virial radius.

One important feature of the COS-Halos O vi measurements (Tumlinson et al. 2011a) is the bimodality in the presence of O vi absorption, with detections around SF galaxies and only upper limits in passive galaxies with $\mathrm{sSFR}\lesssim 4\times {10}^{-12}\,{\mathrm{yr}}^{-1}$. Our model framework does not directly relate the O vi column to the star formation rate (SFR) in the galaxy, but it is consistent with several possible explanations for such a relation. One option is that since the O vi-bearing gas at $T\sim 3\times {10}^{5}$ K has a high cooling efficiency, it requires energy input to prevent it from cooling, as discussed above. In our model, we assume that the radiative losses of the CGM are balanced, and we have shown that SNe, as driven by star formation, can indeed offset these losses. Another scenario, explored by Oppenheimer et al. (2016) using the EAGLE simulations, is that both the SFR and the O vi are independently correlated to the halo (and stellar) mass. In this case, the halo mass with the peak SFR happens to be the same as the halo mass at which the CGM temperature corresponds to the O vi CIE peak. Similarly, in our model, the gas temperature at the outer boundary is related to the halo mass and size (see Equation (13)). In either scenario, (a) halos hosting galaxies with low star formation or (b) higher-mass halos may have hotter coronae with lower cooling efficiencies and longer cooling times. These coronae may be detectable through UV and X-ray absorption of higher metal ions.

Blitz & Robishaw (2000) estimated the CGM density needed to explain the observed dearth of gas in MW dwarf satellite galaxies and found a mean value of $\left\langle {n}_{{\rm{H}}}\right\rangle \approx 2.5\times {10}^{-5}$ cm⁻³ inside 250 kpc. Grcevich & Putman (2009) performed a similar analysis for satellites at distances of $50\mbox{--}100\,\mathrm{kpc}$ and found densities around 10⁻⁴ cm⁻³. Salem et al. (2015) used simulations to reproduce the distribution of ram pressure–stripped gas around the LMC (at a distance of 50 kpc) and estimated a coronal gas density of ${1.10}_{-0.45}^{+0.44}\times {10}^{-4}$ cm⁻³. In our fiducial model, the mean density of warm/hot gas inside 250 kpc is ${\left\langle {n}_{{\rm{H}}}\right\rangle }_{250\mathrm{kpc}}\approx 2.0\times {10}^{-5}$ cm⁻³. The gas densities at $50\mbox{--}100\,\mathrm{kpc}$ in our model are in the range $\sim (0.4\mbox{--}0.7)\times {10}^{-4}$ cm⁻³, a factor of ∼2 lower than the values estimated by Grcevich & Putman (2009) and Salem et al. (2015). They are also lower by a factor of ∼2 than the densities in our FSM17 isothermal model. We note that estimates from ram pressure–stripped systems may be biased toward the denser regions of the corona.

Manchester et al. (2006) presented dispersion measure (DM) measurements to pulsars in the LMC. Anderson & Bregman (2010) discussed these and estimated an upper limit of $\mathrm{DM}\leqslant 23$ cm⁻³ pc for the CGM component. Prochaska & Zheng (2019) used the same observations and estimated $\mathrm{DM}=23\pm 10$ cm⁻³ pc. In our model, the computed DM in the corona to the LMC is $\mathrm{DM}=8.8$ cm⁻³ pc, consistent with the Anderson & Bregman (2010) upper limit.

5.1.1. O vii and O viii Absorption

5.1.2. X-Ray Emission: O vii/O viii Lines and 0.4–2.0 keV Band

5.2.1. O vi and N v Absorption

For the O vi absorption data, we combine two sets of observations. The first are measurements from the COS-Halos survey described in FSM17,¹⁴ probing impact parameters of $h\lesssim 0.6$ ${r}_{\mathrm{vir}}$. The second set consists of measurements from the eCGM survey, presented by Johnson et al. (2015) and extending out to 5–10 virial radii of the observed galaxies. Beyond ${r}_{\mathrm{vir}}$, Johnson et al. (2015) reported mostly upper limits for the O vi column densities, typically below 10¹³ cm⁻² (see Figure 3 there). Since we aim to model MW-like galaxies, we select from this sample the isolated SF galaxies with stellar masses above $\sim 3\times {10}^{9}$ ${M}_{\odot }$, similar to the SF galaxies in the COS-Halos sample. This results in 18 measurements, and the combined data set (COS-Halos and eCGM) is shown in Figure 10 by blue symbols, with measured columns as squares and upper limits as filled triangles.

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The O vi data can be well approximated by a simple step function. Within approximately the virial radius ($h\lesssim {r}_{\mathrm{vir}}$), the profile is consistent with a constant column density of $\sim 4\times {10}^{14}$ cm⁻². At $h\sim {r}_{\mathrm{vir}}$, the column density drops sharply, with only nondetections at larger impact parameters. For $3\,{r}_{\mathrm{vir}}\lt h\lt 7$ ${r}_{\mathrm{vir}}$, the median upper limit is $\sim 7\times {10}^{12}$ cm⁻², a factor of ∼50 lower than the typical column density measured by COS-Halos. For a clearer comparison to our model, we bin the individual measurements in radius in logarithmic intervals, taking the median column density in each bin and estimating the error as the scatter. The binned data are shown by the magenta markers, with the marker size proportional to the number of objects in the bin. The O vi column density profile for our model is the blue solid curve in Figure 10, consistent with the binned data points at $h/{r}_{\mathrm{vir}}\gt 0.1$. The CGM distribution in our model ends at ${r}_{\mathrm{CGM}}=1.1{r}_{\mathrm{vir}}$, chosen to be consistent with the few O vi detections at $h\sim 1.1{r}_{\mathrm{vir}}$ and the nondetections at larger impact parameters.

In the COS-Halos sample, Werk et al. (2014) searched for N v absorption and reported upper limits for most sight lines. For the SF galaxies in the sample, 20 out of 24 sight lines have upper limits for the N v column densities. The COS-Halos N v data are shown in Figure 10 by green symbols, with the nondetections as open triangles and measured columns as filled circles. The median upper limit on the N v column density is $5\times {10}^{13}$ cm⁻², with a scatter of $\lt 0.3$ dex. The cyan dashed curve shows the N v column density profile in our fiducial model. The profile is almost constant with impact parameter, with ${N}_{{\rm{N}}{\rm{V}}}\approx {10}^{13}$ cm⁻², consistent with the measured upper limits. Thus, our model predicts that the N v column densities in the CGM of MW-like galaxies at low redshift are a factor of $\sim 3\mbox{--}10$ below the current upper limits.

The four sight lines with detected N v absorption have column densities between ∼0.5 and $\sim 1.5\times {10}^{14}$ cm⁻². Three of the galaxies associated with these sight lines have stellar masses below $1.5\times {10}^{10}$ ${M}_{\odot }$. This is a factor of 3–4 lower than the MW stellar mass and below the median stellar mass of the COS-Halos SF subsample, ${M}_{* }\approx 2\times {10}^{10}$ ${M}_{\odot }$. In CIE, the N v ion fraction peaks at $\sim 2\times {10}^{5}$ K. These galaxies may have lower halo masses and virial temperatures closer to this value than our fiducial model, leading to an increase in the N v column. The sight line associated with the fourth galaxy has an impact parameter of $h/{r}_{\mathrm{vir}}\sim 0.15$, and the detected absorption may also be contaminated by gas associated with the galactic disk.

5.2.2. X-Ray Emission

The O vii and O viii absorption at $z\sim 0$ has been measured in the X-ray-brightest QSOs with $\sim 30\mbox{--}40$ O vii detections and a handful of sight lines with O viii (Bregman & Lloyd-Davies 2007; Gupta et al. 2012; Miller & Bregman 2013). Fang et al. (2015) searched for a correlation of the O vii column density with Galactic latitude or longitude and found that existing data are consistent with a constant column density profile. However, current absorption observations in the X-ray often have significant uncertainties due to limited sensitivity and spectral resolution. Future X-ray observatories will provide measurements for a larger number of sight lines with higher accuracy (Kaastra et al. 2013; Smith et al. 2016; The Lynx Team 2018), and we calculate the O vii/O viii column distributions in our model to be tested by these observations.

We plot the predicted O vii and O viii column densities in the left panel of Figure 11. Since our model is spherically symmetric, for an observer inside the Galaxy, the column densities, as well as other quantities, are a function only of the angle, ${\theta }_{\mathrm{GC}}$, from the GC. As described in Sections 3.3 and 5.1.1, the O vii ion is abundant at all radii in the CGM, and its half-column length scale, ${L}_{s}\sim 30\,\mathrm{kpc}$, is relatively large compared to ${R}_{0}$. Thus, for an observer at $r\sim 10\,\mathrm{kpc}$, the O vii column density (green curve) is almost constant with ${\theta }_{\mathrm{GC}}$, consistent with current observations (Fang et al. 2015). The O viii ion, on the other hand, is formed mostly in the central part of the CGM, and its length scale is smaller (∼10 kpc). Thus, the column density at small ${\theta }_{\mathrm{GC}}$, with $6\times {10}^{15}$ cm⁻², is higher by a factor of $\sim 2\mbox{--}3$ than at large angles from the center (red curve).

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The DM can provide a strong constraint on the total gas column, since it is independent of the gas metallicity. Today, the DM has been measured for pulsars in the LMC/SMC at a distance of ∼50 kpc (Crawford et al. 2001; Manchester et al. 2006; Ridley et al. 2013). Upcoming facilities (e.g., LOFAR and SKA; van Leeuwen & Stappers 2010; Keane et al. 2015) with higher sensitivities may be able to find pulsars in other, more distant satellites of the MW and measure their DMs. In the left panel of Figure 12, we show the DM in our model as a function of the angle from the GC for distances of d = 50, 150, and 250 kpc from the GC (solid black, blue, and red curves, respectively). The magenta circle marks the LMC at ${\theta }_{\mathrm{GC}}=81^\circ$ with 8.8 cm⁻³ pc. Future DM measurements for extragalactic sources (FRBs, for example) may provide constraints on the total DM of the MW CGM. In our model, the contribution from $r\gt 250\,\mathrm{kpc}$ is small, and integration out to ${r}_{\mathrm{CGM}}$ gives values of DM $\sim 13\mbox{--}21$ cm⁻² pc (dashed black curve), close to the values at 250 kpc.

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Prochaska & Zheng (2019) estimated the DM of the MW CGM at $\sim 50\mbox{--}80$ cm⁻³ pc, integrating to the virial radius. However, this results from models with a large CGM mass,¹⁵ and hence gas density. Their CGM mass is a factor of ∼3 higher than in our model, and scaling down their values for the DM by the same factor gives 17–27 cm⁻³ pc. This is similar to the range in our model when integrated to ${r}_{\mathrm{vir}}$, as shown by the solid red curve in Figure 12. This demonstrates the usefulness of (accurate) DM measurements to constrain the gas density and total mass in the CGM.

6.2.1. UV and X-Ray Absorption

For the MW, UV and X-ray absorption from hot gas has been detected. However, in the X-ray, absorption observations at $z\sim 0$ lack kinematics due to the limited spectral resolution of current instrumentation. In the UV, the detected absorption lines are spectrally resolved, and their kinematics are measured. Nevertheless, the exact location of the absorbing gas is still unclear due to the complex dynamics of the disk–CGM interface (Zheng et al. 2015, 2019; Martin et al. 2019).

Measurements of O vii and O viii absorption in other galaxies (similar to the COS-Halos O vi observations) will better determine the extent of the hot CGM and be more sensitive to low surface density gas compared to emission observations. In the UV, ions such as N v, O vi, Ne viii, and Mg x probe different gas temperatures and can be helpful in constraining the CGM properties. We use our model to predict the column densities for such future observations (Kaastra et al. 2013; The LUVOIR Team 2018). We present the column density profiles for an external observer both as a function of the physical impact parameter and normalized to ${r}_{\mathrm{vir}}$.

The right panel of Figure 11 shows the column density profiles of O vii and O viii versus impact parameter h in our fiducial model (green and red curves, respectively). As discussed above, the O vii ion fraction is high and almost constant across the wide range of temperatures in our model, and the resulting O vii column density profile is extended. It is well fit by an exponential profile, $\mathrm{log}({N}_{{\rm{O}}{\rm{VII}}})\propto \left(-h/{L}_{N}\right)$, with a scale of ${L}_{N}\sim 0.63$ ${r}_{\mathrm{vir}}$ set by the metallicity gradient and the gas density profile. The O viii column (red), on the other hand, has a two-part profile. In the inner regions ($h\lt 25$ kpc), the O viii ion fraction is controlled by collisional ionization and decreases rapidly with temperature. This gives a column density profile that has a strong dependence on the impact parameter, with $N\sim 7\times {10}^{15}$ cm⁻² at 10 kpc. In the outer part, the O viii fraction is set by photoionization and increases with radius between 30 kpc and ${r}_{\mathrm{CGM}}$ (see Figure 6). The resulting column density profile in the outer part is well fit by an exponential function with ${L}_{N}\approx {r}_{\mathrm{vir}}$, flatter than the O vii.

Nicastro et al. (2018, hereafter N18) reported the discovery of O vii absorption in the warm/hot intergalactic material (WHIM). They presented measurements of total oxygen column densities for two absorbers with ${7.8}_{-2.4}^{+3.9}\times {10}^{15}$ and ${4.4}_{-2.0}^{+2.4}\times {10}^{15}$ cm⁻² at z = 0.434 and 0.355, respectively. They searched for possible associations of these absorbers with galaxies, and for the first system, they found a spiral galaxy at a similar redshift and a projected distance of 129 kpc (although see Johnson et al. 2019, suggesting that this absorption is associated with the blazar environment). Assuming this galaxy is indeed associated with the absorber and is similar to the MW, we can compare the measured column to our model. The total oxygen column density from N18 is shown by the black circle in the right panel of Figure 11 (including the 1σ errors reported by the authors). The black curve shows the total oxygen column density in our fiducial model. At an impact parameter of h = 129 kpc, our model predicts ${N}_{{\rm{O}}}=3.0\times {10}^{15}$ cm⁻², dominated by the O vii ion, with ${N}_{{\rm{O}}{\rm{VII}}}=2.3\times {10}^{15}$ cm⁻². This suggests that a nonnegligible fraction of the observed absorption could originate in the warm/hot CGM of the galaxy adjacent to the line of sight, rather than the IGM. Information regarding the stellar or total mass of this galaxy will allow for scaling the impact parameter to the virial radius and performing a better comparison to our model. Furthermore, separating the CGM contribution from the total column will allow a better estimate of the IGM properties. The closest galaxy to the second absorber found by N18 is at a projected distance of 633 kpc, and a similar association with the CGM is less likely.

Figure 13 shows the column densities of several other metal ions observable in the UV. We select N v, O vi, Ne viii, and Mg x—ions present in gas at temperatures between $\sim 2\times {10}^{5}$ and $1.2\times {10}^{6}$ K. First, the N v and O vi profiles (dashed cyan and solid blue curves, respectively) are identical to those presented in Figure 10. As discussed in Section 5.2.1, the O vi and N v ions are abundant mainly in the outer parts of the corona (where ${T}_{\mathrm{gas}}$ is low), resulting in flat column density profiles. The COS-Halos N v absorption measurements give upper limits for a large fraction of the observed sight lines. We predict that the actual column densities are $\sim (0.5\mbox{--}1.0)\,\times {10}^{13}$ cm⁻², a factor of 3–10 below the existing upper limits. Figure 14 shows the column density profiles of several metal ions, normalized by the OVI column density profile. These curves are independent of the mean enrichment of a specific galaxy, and can be useful for comparing to CGM absorption measurements of other galaxies. However, since various ions form at different radii in the halo, the profiles in Figure 14 do depend on the shape of the metallicity profile that we adopt in our model, given by Equation (8).

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The Ne viii and Mg x ions (solid black and magenta curves) probe hotter gas, at $T\sim {10}^{6}$ K, and their ion fractions peak at smaller radii. Thus, their column density profiles have a two-part structure, with high columns at small impact parameters and lower values at larger (projected) distances. Current instrumentation limits observations of Ne viii to $0.5\lt z\lt 1.0$ (Meiring et al. 2013; Hussain et al. 2015; Burchett et al. 2019). To compare current observations with this work, we can assume that the halo and CGM properties of these higher-redshift galaxies are not very different from the MW/COS-Halos galaxies. Our model then predicts that the column density in the central part of the profile, controlled by collisional ionization of Ne viii, will not change significantly with redshift. In the outer part of the corona, Ne viii is created by the MGRF, and for a field intensity higher by a factor of 3–5, the column density at large impact parameters may be higher by a similar factor. Current detections of Mg x absorption are rare and at higher redshifts than our model, $z\gt 1.0$ (Qu & Bregman 2016).

6.2.2. X-Ray Emission and DM

6.2.3. . SZ Effect

We calculate the spatially resolved SZ signal through the corona at an impact parameter h as

$$\begin{eqnarray}&&y(h)=\displaystyle \frac{{\sigma }_{T}}{{m}_{e}{c}^{2}}\int {P}_{{\rm{e}},\mathrm{th}}(r){dz}=\displaystyle \frac{{\sigma }_{T}{k}_{{\rm{B}}}}{{m}_{e}{c}^{2}}\int {n}_{{\rm{e}}}(r){T}_{\mathrm{th}}(r){dz},\end{eqnarray} \tag{ 30 }$$

where ${P}_{{\rm{e}},\mathrm{th}}$ is the electron (thermal) pressure, ${n}_{{\rm{e}}}$ is the electron density, and dz is the element along a line of sight. The resulting y-parameter is shown in Figure 15, and the profile decreases from $\sim {10}^{-8}$ at small impact parameters to $\sim (2\mbox{--}3)\times {10}^{-10}$ at ${r}_{\mathrm{vir}}$. Current CMB observations do not have this sensitivity, and this prediction can be compared to future spatially resolved CMB measurements of galactic halos (see Singh et al. 2015).

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The Planck Collaboration et al. (2013, hereafter P13) searched for the SZ signal from gas in galaxies by stacking CMB measurements of the locally brightest galaxies. They reported the Comptonization parameter normalized to a distance of 500 Mpc, giving the intrinsic integrated SZ signal and defined as

$$\begin{eqnarray}&&{\tilde{Y}}_{500}\equiv \displaystyle \frac{{\sigma }_{T}}{{m}_{e}{c}^{2}}\displaystyle \frac{{E}^{-2/3}(z)}{{\left(500\mathrm{Mpc}\right)}^{2}}{\int }^{{R}_{500}}{P}_{{\rm{e}},\mathrm{th}}{dV},\end{eqnarray} \tag{ 31 }$$

where ${E}^{2}(z)={{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}$. The signal is calculated out to R₅₀₀ of the halo, and ${\tilde{Y}}_{500}$ is usually reported in square arcminutes. We calculate it at z = 0, and $E(z)=1$.

A signal from systems with stellar masses above $\sim {10}^{11}$ ${M}_{\odot }$ was detected by P13. For the lowest-mass bin with a $3\sigma$ detection, ${M}_{* }=2\times {10}^{11}$ ${M}_{\odot }$ (${M}_{500}\sim 2\times {10}^{13}$ ${M}_{\odot }$), they reported ${\tilde{Y}}_{500}\approx 5\times {10}^{-6}\,{\mathrm{arcmin}}^{2}$. To estimate the signal from an MW-mass galaxy,we can use the ${\tilde{Y}}_{500}-{M}_{500}$ relation, usually fit by ${\tilde{Y}}_{500}\propto {M}_{500}^{{a}_{M}}$. In their analysis, P13 adopted the slope predicted by the self-similar solution for the gas distribution in a halo of ${a}_{M}=5/3$. Using this value to calculate the SZ signal for an MW-mass galaxy with ${M}_{500}\,\sim 7\times {10}^{11}$ ${M}_{\odot }$ gives ${\tilde{Y}}_{500}\sim 2\times {10}^{-8}\,{\mathrm{arcmin}}^{2}$. However, P13 noted that a single power law is not a formally acceptable fit to the measured ${\tilde{Y}}_{500}-{M}_{500}$ relation. This may be a result of the gas distributions in galaxies differing from those in clusters (see also Bregman et al. 2018). Thus, the actual SZ signal for MW-mass galaxies may be different from the extrapolated value. For the MW, ${R}_{500}=135\,\mathrm{kpc}$, and in our fiducial model, ${\tilde{Y}}_{500}=0.5\times {10}^{-8}\,{\mathrm{arcmin}}^{2}$.

The angular resolution of the Planck maps used in the P13 stacking analysis is $10^{\prime}$ (FWHM; see Section 5.1 there), and the MW R₅₀₀ will not be resolved at distances above ∼50 Mpc. For spatially unresolved CMB observations, we integrate the SZ signal in our model out to ${r}_{\mathrm{CGM}}$ and get $\tilde{Y}=1.2\times {10}^{-8}\,{\mathrm{arcmin}}^{2}$. This is similar to the estimate by Singh et al. (2015) of ${\tilde{Y}}_{500}\sim {10}^{-8}$ arcmin² for the warm, O vi-bearing CGM.

The column density of ion i at an impact parameter h in a spherically symmetric halo is

$$\begin{eqnarray}&&{N}_{i}(h)=2{A}_{{\rm{i}}}{\int }_{0}^{z}{n}_{{\rm{H}}}(r)Z^{\prime} (r){f}_{V}(r){f}_{\mathrm{ion},{\rm{i}}}(r){dz}^{\prime} ,\end{eqnarray} \tag{ A1 }$$

where ${r}^{2}={h}^{2}+{z}^{2}$, ${A}_{{\rm{i}}}$ is the solar abundance of the element corresponding to ion i, $Z^{\prime}$ is the metallicity relative to solar, f_V is the volume-filling factor of the gas containing ion i, and ${f}_{\mathrm{ion},{\rm{i}}}$ is the ion fraction. We assume a power-law variation of the density, ${n}_{{\rm{H}}}\propto {r}^{-{a}_{n}}$; metallicity, $Z^{\prime} \propto {r}^{-{a}_{Z}}$; filling factor, ${f}_{V}\propto {r}^{-{a}_{V}}$; and ion fraction, ${f}_{\mathrm{ion},{\rm{i}}}\propto {r}^{-{a}_{f}}$. We then have

$$\begin{eqnarray}&&{N}_{i}(h)=2{A}_{{\rm{i}}}{f}_{\mathrm{ion},{\rm{i}}}(h){n}_{{\rm{H}}}(h)Z^{\prime} (h){f}_{{\rm{V}}}(h){\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{{\left(r/h\right)}^{a}},\end{eqnarray} \tag{ A2 }$$

where ${n}_{{\rm{H}}}(h)={n}_{{\rm{H}}}(r=h)$, etc., and $a={a}_{n}+{a}_{Z}+{a}_{V}+{a}_{f}$ ($a\gt 0)$. Let

$$\begin{eqnarray}&&y^{\prime} \equiv \displaystyle \frac{z^{\prime} }{h}={\left(\displaystyle \frac{{r}^{2}}{{h}^{2}}-1\right)}^{1/2},\end{eqnarray} \tag{ A3 }$$

and let R be the virial radius of the Galaxy, close to the outer radius of the CGM. We then define

$$\begin{eqnarray}&&{I}_{a}(y)\equiv \displaystyle \frac{1}{R}{\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{{\left(r/h\right)}^{a}}=\displaystyle \frac{1}{{\left(1+{y}^{2}\right)}^{1/2}}{\int }_{0}^{y}\displaystyle \frac{{dy}^{\prime} }{{\left(1+y{{\prime} }^{2}\right)}^{a/2}}\end{eqnarray} \tag{ A4 }$$

and get

$$\begin{eqnarray}&&{N}_{i}(h)=2{A}_{{\rm{i}}}{f}_{\mathrm{ion},{\rm{i}}}(h){n}_{{\rm{H}}}(h)Z^{\prime} (h){f}_{{\rm{V}}}(h){{RI}}_{a}.\end{eqnarray} \tag{ A5 }$$

If we restrict our attention to normalized impact parameters in the range $0.3\lt h/R\lt 0.9$, which contains most of the COS-Halos measurements (see Figure 10), then for a between 1 and 2, ${I}_{a}=0.50\pm 0.13$ dex (or 0.50 ± 0.18 dex for $a=0.5\mbox{--}2.5$).

Let the rate of radiative net cooling per unit volume be ${n}_{{\rm{e}}}{n}_{{\rm{H}}}{\rm{\Lambda }}$. We assume that the gas is irradiated by the MGRF. The cooling function, Λ, is then a function of the gas density, temperature, and metallicity (see Section 4 and Gnat 2017). The isochoric gas cooling rate is then

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}=\displaystyle \frac{3{{nk}}_{{\rm{B}}}T}{2{n}_{{\rm{e}}}{n}_{{\rm{H}}}{\rm{\Lambda }}(T,n,Z)},\end{eqnarray} \tag{ A6 }$$

where we have adopted ${n}_{\mathrm{He}}={n}_{{\rm{H}}}/12$ and assumed that the gas is fully ionized. For the metallicity scaling, ${\rm{\Lambda }}=Z^{\prime} {{\rm{\Lambda }}}_{\odot }$, we neglect cooling due to H and He, so this an upper limit on the cooling time. Inserting the expression for ${n}_{{\rm{H}}}(h)Z^{\prime} (h)$ from Equation (A5), we get

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}(h)=5.8{A}_{{\rm{i}}}{f}_{{\rm{V}}}(h)\left[\displaystyle \frac{{k}_{{\rm{B}}}T(h){f}_{\mathrm{ion},{\rm{i}}}(h)}{{{\rm{\Lambda }}}_{\odot }(T,n)}\right]\displaystyle \frac{{{RI}}_{a}}{{N}_{i}(h)}.\end{eqnarray} \tag{ A7 }$$

In this expression, the uncertain metallicity $Z$ does not appear, and the cooling time is inversely proportional to the observable column density.

We now apply this to O vi. We assume that the warm/hot gas is volume-filling, so that ${f}_{{\rm{V}}}=1$ and ${a}_{{\rm{V}}}=0$. This gives an upper limit for the cooling time, consistent with the rest of our analysis here. The filling factor of the warm/hot, O vi-bearing gas in our model is unity. The solar abundance of oxygen is ${A}_{{\rm{O}}}=4.9\times {10}^{-4}$, and as we estimated above, ${I}_{a}\approx 0.50$. For our estimate here, we take R = 260 kpc, the median virial radius of the COS-Halos SF galaxies, and close to the MW virial radius in our model (see Section 3). Given the shape of the cooling function and the O vi ion fraction in the density–temperature space, the expression ${k}_{{\rm{B}}}{{Tf}}_{\mathrm{ion},{\rm{i}}}/{{\rm{\Lambda }}}_{\odot }$ is bound from above for gas at $T\gt {10}^{5}$ K. For the HM12 MRGF at z = 0.2, ${k}_{{\rm{B}}}{Tf}_{{\rm{i}}{\rm{o}}{\rm{n}},{\rm{O}}{\rm{V}}{\rm{I}}}/{{\rm{\Lambda }}}_{\odot }\leqslant 4.6\times {10}^{10}\,{\rm{s}}\,{{\rm{c}}{\rm{m}}}^{-3}\,$, and the maximum occurs at $T\sim 3.5\times {10}^{5}$ K at densities above ${n}_{{\rm{H}}}\geqslant {10}^{-4}$ cm⁻³, where the O vi is in CIE and ${f}_{\mathrm{ion},\mathrm{OVI}}$ is maximal (see Section 3.2). We insert this value into Equation (A7) to obtain a model-independent upper limit for the cooling time at r = h:

$$\begin{eqnarray}&&{t}_{\mathrm{cool}}(r=h)\lesssim 5.6\,\mathrm{Gyr}\,\left(\displaystyle \frac{R}{260\,\mathrm{kpc}}\right){\left(\displaystyle \frac{{N}_{{\rm{O}}{\rm{VI}}}(h)}{3\times {10}^{14}{\mathrm{cm}}^{-2}}\right)}^{-1}.\end{eqnarray} \tag{ A8 }$$

This approximation is valid for $0.3\lt h/R\lt 0.9$ through ${N}_{{\rm{O}}{\rm{VI}}}(h)$, and we scaled the column density to the value measured in COS-Halos at $h/R\sim 0.6$ (see Figure 10).

The dynamical time used by Voit et al. (2017) is $\sqrt{2r/g(r)}$, where g(r) is the acceleration due to gravity. Scaling gives

$$\begin{eqnarray}&&{t}_{\mathrm{dyn}}(r)=2.8\,\mathrm{Gyr}{\left(\displaystyle \frac{r}{260\mathrm{kpc}}\right)}^{3/2}{\left(\displaystyle \frac{M(r)}{{10}^{12}{M}_{\odot }}\right)}^{-1/2}.\end{eqnarray} \tag{ A9 }$$

We can fit the Klypin MW-mass profile at large radii, where it is approximately an NFW profile, with a power law, giving

$$\begin{eqnarray}&&M(r)\approx {10}^{12}\,{M}_{\odot }\,{\left(\displaystyle \frac{r}{260\mathrm{kpc}}\right)}^{0.56}(130\,\mathrm{kpc}\lt r\lt 260\,\mathrm{kpc}).\end{eqnarray} \tag{ A10 }$$

Inserting this into Equation (A9) results in

$$\begin{eqnarray}&&{t}_{\mathrm{dyn}}(r)\approx 2.8\,\mathrm{Gyr}\,{\left(\displaystyle \frac{r}{260\mathrm{kpc}}\right)}^{1.22}.\end{eqnarray} \tag{ A11 }$$

We note that, unlike our expression for the cooling time upper limit, this approximation for ${t}_{\mathrm{dyn}}$ is valid all the way out to R.

We can then define $\zeta (r)\equiv {t}_{\mathrm{cool}}(r)/{t}_{\mathrm{dyn}}(r)$ and write the upper limit of this ratio for the typical column of O vi as

$$\begin{eqnarray}&&\zeta (r=h)\lt 2.0\,{\left(\displaystyle \frac{h}{260\mathrm{kpc}}\right)}^{-1.22}{\left(\displaystyle \frac{{N}_{{\rm{O}}{\rm{VI}}}(h)}{3\times {10}^{14}{\mathrm{cm}}^{-2}}\right)}^{-1}.\end{eqnarray} \tag{ A12 }$$

Accounting for the uncertainty factor in the value of I_a gives ratios in the range of 1.5–2.6 for $1\lt a\lt 2$, corresponding to a factor of 1.3 uncertainty; for $0.5\lt a\lt 2.5$, the range is 1.3–3.0, or a factor of 1.5 uncertainty. Our approximation and the derived upper limit are valid for $0.3\lt h/R\lt 0.9$, and the column density we used is measured at $h/R=0.6$, corresponding to h = 156 kpc. Inserting this impact parameter, we get an observed upper limit $\zeta \lt 3.7$ (in the range 2.8–4.8 for $1\lt a\lt 2$). This is significantly lower than values of $\zeta \sim 10$ found by Sharma et al. (2012b) in simulations and Voit et al. (2017) in observations of galaxy clusters.

To summarize, we find that for warm/hot gas with an O vi column density of $\sim 3\times {10}^{14}$ cm⁻² at large impact parameters, observations set an upper bound $\zeta \lesssim 5$. This limit includes the uncertainty in the underlying ion volume density distribution. It is also independent of the exact gas metallicity, as long as the gas cooling in the relevant temperature range is dominated by metals ($Z^{\prime} \gtrsim 0.1$). A ratio $\zeta \sim 10$ would require O vi columns significantly lower than observed in the CGM of L* galaxies.