THE COS-HALOS SURVEY: PHYSICAL CONDITIONS AND BARYONIC MASS IN THE LOW-REDSHIFT CIRCUMGALACTIC MEDIUM

The dimensionless ionization parameter, U, is defined as the ionizing photon density divided by the total hydrogen number density (neutral + ionized). For our purposes we explore models with log U ranging between −1 and −5, with higher values corresponding to a higher ionized gas fraction and a lower gas density. We derive a mean log U of −2.8 for our sample of 33 absorbers, which corresponds to approximately 99% of the hydrogen being ionized, on average. Our adopted values of log U range from −3.8 to −1.6, corresponding to a range in the neutral gas fraction between approximately 25% and 0.01%. Figure 2 shows log U versus the H i column density for 44 COS-Halos absorbers within 160 kpc of an L ∼ L* galaxy, with the 11 galaxies excluded from our analysis shown for reference as gray ×'s at an arbitrarily chosen value of log U = −1. Upper and lower limits on the measured H i column densities from the COS data are shown, respectively, by left-facing and right-facing arrows. The colored data points (blue = star-forming; red = non-star-forming) for the 33 lines of sight with metal line data show a clear trend of decreasing ionization parameter with increasing H i column density. Lehner et al. (2013) carried out a similar CLOUDY-based analysis for Lyman limit systems with log N_H i > 16.0, and found a mean log U of −3.3 ± 0.6. When we consider our absorption line systems at similar H i column densities, we also find a mean log U of −3.3.

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To assess the strength of the correlation between log U and H i column density, we perform a statistical analysis using the ASURV software package Rev 1.2 (LaValley et al. 1992), which implements the methods presented in Feigelson & Nelson (1985), and Isobe et al. (1986). We include censored data points (i.e., lower limits) in the H i column density. We implicitly assume that the limits are random with respect to the galaxies. Given the range of impact parameters sampled and that the quasars were selected without any knowledge of the absorption, this assumption should hold. We exclude the 11 galaxies with no H i and/or metal-line detections from this analysis, since those data points exhibit censoring in H i column density and log U, which would make any fit unconstrained. We reject the null hypothesis with a probability of 99.994% by performing a Kendall Tau test of censored bivariate data in this parameter space. Thus, log U and log N_H i are very likely anti-correlated. To derive the best linear fit to the observed correlation, we perform a linear regression analysis for 1000 trials with the log U values randomly distributed along the range of allowed log U values for each absorber (shown by the gray lines in Figure 2). The best-fit, which includes H i column density lower limits, is shown as a solid blue line, and is given by:

$$\begin{equation} \log U = (-0.24\pm 0.06) \log {N}_{{\rm H\,\scriptsize{I}}} + (1.3\pm 0.5). \end{equation} \tag{ 2 }$$

The dotted lines show 1σ confidence intervals of the fit, and the standard error of the regression is 0.38.

Interestingly, this anti-correlation between log U and log N_H i is qualitatively consistent with predictions for photoionized clouds in (or not far from) local hydrostatic equilibrium. Schaye (2001) argues that over-dense absorbers, such as those we may observe in the CGM, have sizes of the order of the local Jeans length, regardless of whether the cloud as a whole is in hydrostatic equilibrium. A relation between neutral hydrogen column density and the characteristic volume density naturally arises from this requirement (Equation (8) of Schaye 2001) such that as N_H i increases, n_H also increases (equivalently, log U decreases; see also Prochaska et al. 2004). We explore several hydrostatic solutions in Section 5.3.

To compare log U to the low ionization state metal line absorption, we derive the quantity N_Low for each system, defined as follows (Werk et al. 2013): (1) N_Low = N_Si ii, if the Si ii measurement is a value or lower limit; (2) N_Low = N_Mg ii if the Si ii measurement is an upper limit (or there is none recorded) and an Mg ii measurement exists. We choose Si ii as the primary low ion because it has multiple transitions in the far-UV bandpass with a range of oscillator strengths yielding more reliable column density estimates. The Mg ii doublet, meanwhile, offers more sensitive upper limits. Because we measure N_Si ii ≈ N_Mg ii in cases where both are measured, we apply no offset when adopting one versus the other. In all, there are roughly half of the systems in each category. Figure 3 shows log U versus N_Low for our sample, and bears a high degree of similarity to Figure 2 such that higher values of N_Low exhibit lower values of log U. Werk et al. (2013) have shown that N_Low is significantly coupled to H i column density, with low-ion detections essentially requiring a non-negligible opacity at the H i Lyman limit. Together, Figures 2 and 3 show that this observed trend follows from the ionization parameter of the gas being higher at lower low-ion and H i column densities. We explore the physical significance of this correlation in Section 5.

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Figure 4 shows log U versus R/R_vir for the same sample. Despite the large scatter, Figure 4 shows that ionization parameter and impact parameter are positively correlated. A Kendall-Tau test of 1000 trials with the log U values randomly distributed along the range of allowed values rejects the null hypothesis with a probability of 97%, indicating a positive correlation is present. Based on a linear regression analysis (as implemented in ASURV), we find that the best power-law fit to this correlation is:

$$\begin{equation} U = 0.006\pm 0.003 (R/R_{\rm vir})^{0.8\pm 0.3} \end{equation} \tag{ 3 }$$

with a combined standard deviation of ∼0.5 (shown in blue, with 1σ confidence intervals shown). Thus, the CGM is more highly ionized farther from its host galaxy. We discuss the extent to which this trend is the result of a declining gas density gradient, in Section 4.1.

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As described throughout this work and in detail in the Appendix, our constraints on the gas metallicity are generally poor owing to the line saturation of H i Lyα and other Lyman series lines in our COS data. Nonetheless, for 11 COS-Halos absorbers with well-constrained H i column densities, we are able to estimate the gas metallicity to better than ±0.2 dex. Our values of [X/H] range from −1.5 to solar, and are determined for galaxies with either log N_H i < 16.5 cm⁻² where Lyman series lines do not saturate in our COS data or log N_H i > 18.5 where the presence of damping wings on Lyman series absorption lines allows for an estimate of the H i column density from Voigt profile fits (Tumlinson et al. 2013). The 11 systems that do not show absorption from metal ions do not offer any useful upper limits on the gas metallicity. We discuss the limited constraints implied by metal line non-detections more fully in the next section.

Figure 5 compares our gas metallicity for the CGM of L* galaxies within 160 kpc to the values of Lehner et al. (2013), who analyzed 28 Lyman limit systems and found a bimodal metallicity distribution. We note that the Lehner et al. study examines a range of H i column density where constraints from COS-Halos are the weakest (16.2 < log N_H i < 18.5). A key difference between this study and that of Lehner et al. (2013) is that the COS-Halos target selection is based on galaxy properties, while the Lehner et al. (2013) target selection is based on H i column density selection that allows them to sensitively probe both high and low metallicities. The range of gas metallicities we find is similar to that of the Lyman limit systems examined by Lehner et al. (2013), but we do not have enough data to distinguish any bimodality in metallicity for COS-Halos galaxies. Moreover, we have not analyzed the 11 cases in which no metals are apparent (Q = 1), and it is possible some of those cases would occupy the low-metallicity branch.

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With respect to a metallicity gradient with R, we find no apparent trend between metallicity and impact parameter for the 33 COS-Halos absorbers included in this analysis, albeit very large uncertainties and upper limits. We have performed a survival analysis in this parameter space using the ASURV to test for any hint of a correlation. A Kendall-Tau test, which includes censoring (upper limits) and 1000 trials with the [X/H] values lying along a randomly populated distribution defined by the range of allowed values, reveals a z-value of 0.689 and rejects the null hypothesis with a probability of only 41%. Because the gas metallicity is poorly constrained by our data, we see no evidence of a metallicity gradient with an impact parameter. The best fitting power-law slope is [X/H] ∝ (R/R_vir)^{0.2 ± 0.4}. We explore whether this finding is consistent with theoretical predictions from a primarily wind-fed CGM and with naive expectations based on several possible origins of the CGM in Section 5.

For 7 of the 44 sightlines considered, we have not detected absorption lines other than H i Lyα or Lyβ. Four additional galaxy-halo sightlines are devoid of absorption from any ion, including H i Lyα. In this section, we address whether these "undetected" systems are more likely to exhibit low metallicity and/or low N_H compared to the sample of 33 absorbers for which we can model the gas using CLOUDY. To do this, we examine the typical upper limits to the column densities of several common metal ions in the parameter space of metallicity and N_H. We show that the seven sightlines with H i absorption but no metal absorption can exhibit the full range of physical conditions (ionization parameters, metallicities) for the gas that shows detected metal-ion absorption.

In Figure 6, we plot CLOUDY-derived gas metallicities as a function of the total hydrogen column for the 33 systems showing H i and metal lines. The sizes of the data points are inversely proportional to the errors in their derived quantities, such that larger data points have well-constrained ionization parameters, H i column densities, and/or metallicity. Upper limits to metallicity and lower limits to the total hydrogen column density are shown as diagonal arrows to the bottom right, as the two quantities are degenerate. The filled gray region of this plot showcases the region of this parameter space that is ruled out by the typical 2σ upper limits to the Si iii column density. These six limits have all been scaled to log N_H i = 15.0 to show in this parameter space. Si iii is the most commonly detected metal ion with the best coverage in the COS-Halos data set, and in six of the seven non-detections we can place reliable upper limits on its column density. By comparison, non-detections of the lower ionization state metal ions (e.g., Mg ii, Si ii, and C ii) are completely consistent with the full range of parameter space shown. Other intermediate-ion non-detections, such as N iii and C iii, are only informative in one and two cases of non-detections. We show the constraints of those upper limits as solid lines of red and blue for C iii and N ii, respectively. The regions not allowed by these two limiting cases would lie above the plotted curves. Next to the ion name we give the number of 2σ upper limits (out of seven possible) that were averaged.

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Figure 6 shows that the allowed metallicities and N_H values (i.e., unshaded area) for the "undetected" systems are largely consistent with the ranges exhibited by the 33 data points from the "detected" systems. Thus, there is no reason to assign them unusual N_H. In general, the 11 systems excluded from this analysis have low H i column densities, log N_H i < 15.0. Based on the relation derived in Section 3.1, we might expect the optically thin gas along these sightlines to have a log U ≈ −2, and thus a total hydrogen column density of log N_H ≈ 10¹⁹ cm⁻². At this value of N_H, the upper limits to the column densities of intermediate ionization state lines for 6/7 of the "undetected" systems are consistent with the full range of metallicity considered, from 0.01 solar to solar. As shown, there is one case where the upper limit on CIII can rule out a gas total hydrogen column and metallicity that most commonly describes our sample ([X/H] = −0.5; log N_H = 19.6—area above the blue contour). In other words, this sightline requires the assumption of a low gas metallicity ([X/H] < −1.0) if the total hydrogen column of the gas is near the median of the detected sample (log N_H = 19.6 cm⁻²).

Other than this one upper limit to N_C iii, the metal ion non-detections can easily exhibit the same metallicity and total hydrogen column density as the gas we do detect. However, the inverse is also true. The 11 systems for which we detect no metal ion absorption can also exhibit lower gas metallicity and/or lower N_H. Essentially, we cannot draw any conclusions about the physical conditions of the gas (or lack thereof) for the 11 "undetected" systems. Hence, we do not have a compelling reason to believe we are biasing our results one way or the other by simply excluding them from our analysis.

The total hydrogen column density, N_H, is simply the sum of the neutral and ionized hydrogen column densities, where we determine the ionized hydrogen column density directly from the derived ionization parameter of the gas. Thus, N_H = N_H i + N_H i/(1−χ), where χ is the ionized gas fraction (mean ≈99%). Table 1 lists both the range of allowed N_H and the adopted N_H. The low value is based upon the H i column density from Tumlinson et al. (2013; typically the AODM column density) and the lowest possible ionization parameter allowed by the photoionization modeling. The high N_H value is based on the preferred H i column density and the highest possible ionization parameter allowed by the photoionization modeling. This high value is still a lower limit because of H i saturation. In the Appendix, we discuss how we determine a "preferred" H i column density. In brief, it is equivalent to the measured AODM column density, with the exception of several lower limits which have been raised to make the gas metallicity consistent with being solar or below. The value for N_H we adopt is calculated using the preferred H i column density and the mean log U for the full range.

Some values of N_H are determined to ±0.2 dex, accounting for the systematic errors in the CLOUDY modeling, uncertainties in the derived column densities, and allowing for ∼0.1 dex uncertainty that arises if the elemental abundance ratios are non-solar. Uncertainty in the slope of the EUVB adds an additional ±0.3 dex of systematic error. Slopes shallower than HM2001 will yield higher ionization parameters, and steeper slopes result in smaller ionization parameters. We describe the details of the error accounting in the Appendix. The mean uncertainty in N_H is ±0.5 dex. For the statistical tests and linear regression analysis described in this section, we sample a random distribution along the full "allowed" range of N_H 1000 times. We "censor" those sightlines for which H i is saturated, and thus include them as lower limits.

Figures 7 and 8 show the total hydrogen column density as a function of impact parameter, and are effectively gas surface-density profiles. There is a 92% chance that log N_H declines with impact parameter, and the implications of this trend are discussed more fully in Section 5. Figure 7 shows this result, with blue and red data points representing star-forming and non-star-forming galaxies, respectively. Figure 8 shows the same results, limited to the modeled systems, with data divided into three bins of R/R_vir, and the best power-law fit with 1σ errors shown in green. We perform a survival analysis on these data by populating a random distribution along a range of allowed hydrogen column density 1000 times, and including censored data points where the H i (and thus N_H) is a lower limit. The best power-law fit from a linear regression analysis is:

$$\begin{equation} N_{\rm H, preferred} = 10^{19.1 \pm 0.5} (R/R_{\rm vir})^{-1.0\pm 0.5}\ \rm cm^{-2}. \end{equation} \tag{ 4 }$$

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We note that without the lower limits in the gas column densities, the null hypothesis is rejected with a confidence of 99%.

Additionally, in Figure 8 we show in the shaded beige area the best fit that now includes the 11 metal line non-detections. We sample the full range of allowed total hydrogen column densities for the seven systems showing H i but no metal lines. The range spans the H i column density on the lower side to the total possible N_H allowed by the 2σ upper limits to the metal-ion column densities on the upper side. For a typical absorber in this sample, the full range of N_H is 10¹⁵ ∼ 10^20.5. For the four systems that show no absorption at all, we include them at a N_H of 0. The best power-law fit, shown by the dashed brown line within the beige area that encompasses the 1σ error to this fit is:

$$\begin{equation} N_{\rm H, low} = 10^{19.1 \pm 1.2} (R/R_{\rm vir})^{-0.4\pm 1.3}\ \rm cm^{-2}. \end{equation} \tag{ 5 }$$

Finally, we determine a total low-ion silicon column density as a function of impact parameter in Figure 9. We calculate log N_Si for each CLOUDY-modeled sightline by applying the ionization corrections to the lowion metal lines we observe, and converting to silicon by assuming solar relative abundances. Since most lines of sight provide good estimates of low-ion metal column densities without saturation, this metal surface density determination is more robust than the total gas surface density determination. Metal surface density is more reliable than metallicity because it does not include uncertain H i column densities. The best power law fit from a linear regression analysis is:

$$\begin{equation} N_{\rm Si} = 10^{13.5 \pm 0.3} (R/R_{\rm vir})^{-0.8\pm 0.3}\ \rm cm^{-2}. \end{equation} \tag{ 6 }$$

Peeples et al. (2014) explore the implications on the total metal mass of the cool CGM that this metal surface density implies.

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Along with ionization parameters and metallicities, the CLOUDY modeling of our COS data allows us to estimate the gas volume density along each sightline, where n_H = Φ/Uc. Here, Φ is the total flux of ionizing photons (∼1.21 × 10⁴ cm⁻² s⁻¹), as defined by the Haardt & Madau (2001) background radiation field from quasars and galaxies, and c is the speed of light. This quantity n_H refers to the volume density of the gas that gives rise to the low-ionization-state metal lines and neutral hydrogen, and is thus not intended to represent the average volume or mass-weighted density of the CGM. The range of allowed n_H directly results from the range of allowed gas ionization parameters. In Figure 10, we show this gas number density, parameterized to the cosmic mean density of hydrogen at z ∼ 0.2 of 3.3 × 10⁻⁷ cm⁻³, as a function of impact parameter scaled to the galaxy virial radius. Reflecting the increase in ionization parameter with R (Figure 4), there is an indication of declining gas volume density with distance from its host galaxy. A Kendall-Tau test rejects the null hypothesis at the 2σ level, and the best power-law fit is:

$$\begin{equation} n_{\rm H}/\langle n_{\rm H} \rangle = 2.2\pm 0.25(R/R_{\rm vir})^{-0.8\pm 0.33}. \end{equation} \tag{ 7 }$$

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We have shown that the photoionized gas in the CGM of L ≈ L* galaxies out to 160 kpc is highly ionized, where the mean neutral gas fraction is ∼1%. The total covering fraction of the gas is 90% ± 4% at N_H i > 10¹⁴ cm⁻² (Thom et al. 2012; Tumlinson et al. 2013; Werk et al. 2013). Here, we estimate the total contribution of the photoionized CGM to the baryonic budget of an L ≈ L* galaxy treating the COS-Halos sightlines as probes of a single "fiducial" galaxy halo. This method leverages the careful, unbiased sample selection of COS-Halos: galaxy-quasar pairs were selected purely on the basis of galaxy properties with no foreknowledge of absorption. Effectively, we have generated the first statistical map of the CGM with 44 individual probes out to 160 kpc that allow us to calculate the mass by exploiting our knowledge of the gas surface density. This method requires no assumptions about volume-filling factors and cloud sizes, and relies only on measured column densities of low-ionization state metal lines and H i, along with CLOUDY-derived ionization parameters based on these quantities. In this sense, it is the first unbiased estimate of the total baryonic budget of the photoionized L ∼ L* CGM, and is independent of models of halo gas density or dark-matter mass.

We make two estimates for the total mass of the CGM based on the CLOUDY modeling: a strict lower limit, and a preferred lower limit. We do so in order to account for the large systematic errors associated with our analysis, discussed more fully in the Appendix. Each calculation is described below, and relies upon converting the total hydrogen column density distribution to a gas surface density distribution by mass. For an additional mass estimate, we calculate the mass of the CGM by estimating the individual cloud sizes (N_H/n_H) and masses indicated by the absorption along each line of sight, and populating a CGM with these clouds to 300 kpc as is observed (Prochaska et al. 2011).

4.2.1. Strict Lower Limit

We base the strict lower limit to the baryonic mass of the photoionized, cool CGM on three very conservative assumptions regarding our data: (1) the AODM H i column density we measure from the COS spectra is the true H i column density, regardless of whether it is saturated in the data or if adopting this value requires assuming a super-solar gas metallicity; (2) the lowest ionization parameter (i.e., highest neutral gas fraction) allowed by the CLOUDY modeling is the true ionization parameter of the gas; and (3) as our observations include sightlines that only lie up to 160 kpc (0.55 R_vir) in projection from the massive host galaxy, we assume the gaseous CGM abruptly "ends" beyond this value. We include the 11 non-detections in this estimate as described in Section 4.1.

The mean total hydrogen column assuming these minimal H i values and ionization parameters is log N_H = 19 cm⁻². The best power-law fit for a gas surface density profile based on these values, truncated to 160 kpc (0.55 R/R_vir), is given by Equation (5). The corresponding gas surface density by mass is 1.4 m_pN_H(r), abbreviated here as Σ_gas(r), where the factor of 1.4 corrects for the presence of helium (the other metals make a negligible contribution to the baryonic mass). It then follows that the total mass is

$$\begin{equation} M_{\rm CGM}^{\rm cool}= \int 2\pi R \Sigma _{\rm gas}(R)\,{dR}. \end{equation} \tag{ 8 }$$

Integrating this equation from 0 to 0.55 R/R_vir, we find a strict lower limit to the mass of the photoionized CGM of 2.1 × 10¹⁰ M_☉.

4.2.2. Preferred Lower Limit

4.2.3. Volume-filling Factor

The median halo mass of our sample of 44 COS-Halos galaxies, based on abundance matching, is 1.6 × 10¹² M_☉ (Moster et al. 2010). Thus, the cosmological baryonic budget of the typical COS-Halos galaxy is approximately 10^11.4 M_☉ (17% of the DM component), with the stellar disk contributing approximately 10^10.6 M_☉, or 14%. The gas in the ISM of these galaxies will vary from very little to nearly as much as the stellar contribution (McGaugh et al. 2010; Martin et al. 2010). We find that the contribution from the photoionized, bound CGM to the total baryonic budget of an L ∼ L* galaxy is at least 25%—at least as much as the total contribution from the stars and gas in the disk of the galaxy.

Studies of warm-hot gas phases of the CGM indicate that 10⁵–10⁷ K gas likely comprises an additional, significant contribution to the baryonic content of galaxy halos (e.g., Tumlinson et al. 2011; Tripp et al. 2011; Anderson et al. 2013; Fox et al. 2013; Meiring et al. 2013). For example, our best CLOUDY models that fit the low ionization state metal absorption lines systemically underestimate the column density of O vi measured from our COS data (mean log N_O vi = 14.5 cm⁻²), as described in Section 3 and shown in the figures in the Appendix. As a result, our best models require that the majority of the observed O vi lie in a separate, more highly ionized gas phase which is consistent with the findings of previous studies. The mass estimates of this more highly ionized phase traced by O vi absorption are complicated by the lack of any additional metal line transitions near the ionization potential of O vi in the COS-Halos data. Tumlinson et al. (2011) estimate a lower limit to be >10⁹ M_☉, based on the maximum possible value for the ionization fraction of O vi (f_O vi < 0.2) and solar metallicity. Peeples et al. (2014) have refined this estimate by considering grids of simple one-dimensional halo models that account for the observed O vi, where the temperature ranges from 10⁴ to 10⁶ K, and the gas surface density profile drops as (R/300 kpc)^α with alpha values of −1 and −2. The typical value for the mass of the O vi-traced CGM is several times higher than the lower limit, and lies near ∼10¹⁰ M_☉. Thus, the O vi gas phase contributes at least 5% of the total baryons in the fiducial COS-Halos galaxy halo. For reference, if the gas is instead assumed to be 0.1 Z_☉, this mass contribution rises accordingly by a factor of 10.

The contribution of a diffuse, X-ray component to the CGM has been subject to debate, with estimates ranging from 10⁹ to 10¹¹ M_☉ (Anderson & Bregman 2010; Gupta et al. 2012; Anderson et al. 2013). Stacked ROSAT images indicate a mass of gas of ∼10⁹ M_☉ of 5 million degree gas within 50 kpc of an L* galaxy (Anderson et al. 2013). Extrapolating this value to 300 kpc, which may or may not be justified, is complicated by the unknown slope of the hot halo gas profile, but will increase the total mass by a factor between 6 and 14. In this estimate, the mass of the X-ray traced CGM is approximately equal to that of the O vi-traced CGM phase.

In contrast, Gupta et al. (2012) argue that the mass content of Milky Way gas at 2 × 10⁶ K out to 160 kpc is greater than 10¹¹ M_☉ based on O vii column densities from XMM Newton data of eight bright AGN and an average emission measure of the soft X-ray background (see also Fang et al. 2013). This result implicitly assumes that the O vii absorption lines and the background emission arise from the same gas phase at the same temperature and metallicity, an assumption that has generated some criticism (Wang & Yao 2012; Fang et al. 2013). Taken at face value, this estimate would mean that the X-ray traced CGM to 300 kpc could comprise at least 50% of the total baryonic budget of the Milky Way.

Finally, recent observations indicate an extensive cool, dusty component of the CGM, possibly fed by a "slow flow" of dust that has coupled with the radiation from massive stars (Zahid et al. 2013). Based on a statistical analysis of reddening using SDSS quasars behind galaxies, Ménard et al. (2010) estimate a total CGM dust mass of ∼5 × 10⁷ M_☉. While this does not contribute substantially to the total baryonic budget of massive galaxies, we note that this mass implies a large fraction of the total metals in the CGM are in a solid, cool phase (Peeples et al. 2014).

Summing the total contributions of all the distinct phases of the CGM, we estimate that the diffuse gas in galaxy halos accounts for at least 35% of the total baryon budget for nearby, L ∼ L* galaxies. This quantity makes up more than half of the baryons purported to be missing (∼60%). Accounting for saturation in the H i column densities used in the cool CGM calculation may raise this contribution by an additional 20%. Thus, the baryonic fraction of L* galaxy halos may be consistent with the cosmological baryon fraction.

We now consider inferences that may be drawn on the nature of the CGM in the context of simple, hydrostatic solutions. Before proceeding, we will review the main characteristics of the CGM revealed by our COS-Halos program and previous works. First, the cool CGM is nearly ubiquitous. The covering fraction f_C of H i gas exceeds 90% for sightlines intersecting 0.55 R_vir (Tumlinson et al. 2013), and the incidence of lower ionization state metals is comparable (Werk et al. 2013). This material, therefore, is pervasive within the dark matter halos of L* galaxies, but we allow that the volume-filling factor f_V may be small (see Section 4.2.3). Second, as emphasized in Section 4.1, the surface density of cool gas is large $N_{\rm H}^{\rm cool}$ > 10¹⁹ cm⁻² at essentially all impact parameters R < R_Vir. Third, the cool CGM is photoionized with T ∼ 10⁴ K. This is indicated by ratios like Si⁺⁺/Si⁺ (Werk et al. 2013) and the detailed photoionization models presented here (see also Stocke et al. 2013). Fourth, the CGM of star-forming galaxies also exhibits a highly ionized phase traced by O vi (Tumlinson et al. 2011). As detailed in our Appendix, this material cannot be reproduced by the photoionization models derived to match the lower ionization states of metals observed. This O vi gas presumably traces another "phase" of the CGM. Lastly, we assume the galaxies under study exist within dark matter halos having T_Vir ≳ 10⁶ K based on the stellar mass estimates (see Section 2).

5.3.1. Single-phase Solutions

The simplest model to consider is a single-phase CGM in hydrostatic equilibrium with the dark matter halo potential. For the latter, we assume an Navarro, Frenk, & White (NFW) profile

$$\begin{equation} \rho _{\rm DM}(r) = \frac{\rho _S r_S^3}{r (r+r_S)^2} \end{equation} \tag{ 9 }$$

defined by a scale radius r_S set by the concentration parameter C_V ≡ r_V/r_S = 13 and with ρ_S set by the halo mass, which we take as M_DM = 10¹² M_☉.

For the baryons, we assume the CGM is an optically thin medium, with metallicity 1/10 solar irradiated by the extragalactic UV background (Haardt & Madau 2001). We allow for a total gas mass M_g as large as Ω_bM_DM/Ω_m but also consider smaller masses parameterized by f_g.

The astrophysical solutions for a baryonic plasma embedded within an NFW potential have been considered many times previously (e.g., Makino et al. 1998; Suto et al. 1998; Capelo et al. 2010), primarily in the context of hot gas around massive elliptical galaxies or the intracluster medium. Our scenario differs in that our fiducial halo is somewhat less massive and, more importantly, we consider a plasma with substantially lower gas temperature.

To simplify the analysis, we make two standard approximations: (1) the baryons do not contribute to the gravitational potential. In the extreme case, M_g may represent as much as Ω_b/Ω_m ≈ 0.2 of the total mass which is a modest contribution; (2) the gas follows a polytropic law $P \sim \rho ^\Gamma$. For an optically thin and photoionized gas, the temperature is relatively insensitive to the gas density. Examining the output of the photoionization models presented in Section 3 for the EUVB background and a 1/10 solar metallicity, we find that $T \propto n_{\rm H}^{1/5}$ for log n_H ≈ −5 to −1. This gives a polytropic index of Γ = 0.8.

We consider first the isothermal case (Γ = 1) with T₀ = 2 × 10⁴ K. Following the formalism of Capelo et al. (2010), the density profile is given by

$$\begin{equation} \rho (r) = \rho _0 \, \exp \left[ \,-\Delta _{\rm NFW} \, \left(\,1 - \frac{\ln (1+r/r_S)}{r/r_S} \, \right) \, \right] \end{equation} \tag{ 10 }$$

with Δ_NFW = −ϕ₀ρ₀/P₀ = −ϕ₀μm_p/kT₀ and the central gravitational potential ϕ₀ ≈ 10⁵ cm² s⁻¹ for our fiducial halo. Therefore, we have Δ_NFW ≈ 1000 (T/10⁴ K)⁻¹ and the density falls off very steeply with radius resulting in a negligible value in the outer halo (r > r_s). We conclude that an isothermal, cool CGM cannot reproduce the observations.

For the polytropic case, we have

$$\begin{equation} \rho (r) = \rho _0 \, \left[ \,1 - \frac{\Gamma -1}{\Gamma } \Delta _{\rm NFW} \, \left(\,1 - \frac{\ln (1+r/r_S)}{r/r_S} \, \right) \, \right] ^{1/\Gamma -1}. \end{equation} \tag{ 11 }$$

This leads to a slightly shallower density profile for Γ = 0.8 but still one where ρ(r_S) ≈ ρ₀/10¹⁰. The outer halo is very nearly a vacuum.

We conclude that a single-phase CGM with T ≈ 10⁴ K in hydrostatic equilibrium with an NFW potential cannot reproduce the observations. We are motivated, therefore, to consider more complex (and realistic) scenarios.

5.3.2. Two-phase Models

Guided by the observations for a wide range of ionization states in halo gas (e.g., Si ii, Mg ii, Si iii, C iv, O vi), it is likely that the medium has multiple phases. In their seminal paper, Mo & Miralda-Escude (1996) introduced a two-phase model composed of cool clouds (T ∼ 10⁴ K) in pressure equilibrium with a more diffuse, hot halo gas (T ∼ 10⁶ K). They presented solutions for the density profile of the hot phase, placed constraints on the masses of the cool clouds, and tracked the dynamics (i.e., infall kinematics) of the cool clumps. In turn, they demonstrated that this two-phase model could reproduce some of the basic observables of halo gas at z < 1.

Other authors have since developed hydrostatic solutions in the context of high velocity clouds of the Milky Way (Sternberg et al. 2002). These treat the EUVB radiation field to model the log N_H i profile of the clouds and also examine higher ionization states of the gas. In all of these models, the cool phase is assumed to have "condensed" out of the hot phase via hydrostatic instabilities. While there is theoretical support for this assumption (Field 1965), other analyses have argued that galactic halos are generally stable to such condensations (Binney et al. 2009; but see McCourt et al. 2012).

In the following, we adopt the formalism of Maller & Bullock (2004, hereafter MB04), who expanded upon the Mo & Miralda-Escude (1996) treatment. Our goal is to examine whether such clumpy, two-phase scenarios reasonably reproduce the gas volume densities and surface densities estimated from our data set.

MB04 assumed that the hot gas originally follows the NFW dark matter profile (Equation (9)) with an inner core (Frenk et al. 1999). Within a characteristic cooling radius r_C, a fraction of the mass takes the form of cool, pressure-supported clouds and the hot gas evolves adiabatically to a new density profile:

$$\begin{equation} \rho _h(x) = \rho _c \left\lbrace 1 + \frac{3.7}{x} \ln (1+x) + \frac{3.7}{C_C} \ln (1+C_C) \right\rbrace ^{3/2} \end{equation} \tag{ 12 }$$

with x = r/r_S and C_C = r_C/r_S. Given this density profile for the hot gas (and a related expression for the temperature which has a small variation), we estimate the cool gas density as

$$\begin{equation} \rho _{\rm cool}(r) = \rho _h(r) \frac{T_h(r)}{T_{\rm cool}(r)} \end{equation} \tag{ 13 }$$

and we adopt T_cool = 2 × 10⁴ K in what follows. All of the remaining variables relate to the assumed properties of the halo. For our fiducial model, we take M_halo = 10¹² M_☉, r_V = 199 kpc, T_halo = 1.3 × 10⁶ K, C_V = 13, and a halo gas metallicity Z_halo = 0.1 Z_☉. Also, we perform the calculation at z = 0 but note the results are similar for any z ≪ 1.

To compare against the measurements along the quasar sightlines which intersect halos at fixed impact parameters R, we must project the gas density profile. Because our analysis is weighted by column density (e.g., we sum all of the gas along each sightline), we calculate a density-weighted value, which closely approximates projection effects:

$$\begin{equation} \rho (R) = \frac{\int \rho ^2(r) ds}{\int \rho (r) ds}. \end{equation} \tag{ 14 }$$

The model is compared against the measurements in Figure 12, with each of our derived n_H values converted to an electron density based on the ionized gas fractions of hydrogen and helium. The best fit to our observations, represented by the shaded light brown area and binned data points, shows electron densities approximately two orders of magnitude below the predicted cool gas electron density profile of MB04. Counter to expectation, the data follow the hot gas electron density profile very well. Figure 12 shows that the inferred n_H values of the CGM lie approximately two orders of magnitude below predictions for a standard, simple two-phase scenario. Thus, in these simplified one- and two-phase cases, we have ruled out static solutions for the cool CGM.

5.3.3. The Failure of the Two-phase Solution

Here, we explore whether the CGM gas masses and densities we measure are consistent with the gas having originated in a wind from the central host galaxy. Several recent cosmological simulations that examine the CGM incorporate various feedback prescriptions, yet all of them indicate that most of the observed absorption in the halos of galaxies is due to gas that is, or was at some point, in a galaxy-scale wind (Shen et al. 2012; Stinson et al. 2012; Joung et al. 2012; Cen 2013; Ford et al. 2013; Hummels et al. 2013). Furthermore, Cen (2013) predict that over half of the gas within 150 kpc is in the cool 10⁴ K phase, consistent with our observations. The idea that winds eject a substantial amount of material into galaxy halos is echoed at least in part by the recent finding that galaxies themselves seem to be missing over half of the metals their stars have produced over the course of their lives (Zahid et al. 2012; Peeples et al. 2014). Toward galaxies themselves, the imprinted signatures of inflows and/or outflows on absorption line profiles have been studied over the last decade for modest samples of individual galaxies (e.g., Heckman et al. 2000; Martin 2005; Rubin et al. 2011), in co-added spectra of many galaxies (e.g., Weiner et al. 2009; Bordoloi et al. 2013), and most recently, for a few hundred galaxies with HST imaging (Martin et al. 2012; Rubin et al. 2013). These studies have shown that the Doppler blueshift of Mg ii of Fe ii absorption (indicative of outflows) in a galaxy spectrum is stronger with higher SFRs, is more common for galaxies that are oriented face-on, and that such outflows have the capacity to transport a significant amount of mass into the CGM and are likely to be highly collimated.

Based on absorption line profiles of Mg ii and Fe ii for 105 individual star-forming galaxies at 0.3 < z < 0.7, Rubin et al. (2013) estimate a mass outflow rate of at least 1 M_☉ yr⁻¹. Assuming the outflow rate remains constant from z ∼ 1 to 0, and that none of the material is re-accreted, this rate implies over 10¹⁰ M_☉ of cool, photoionized material in the CGM of present-day galaxies, consistent with our estimates for the mass of the cool CGM. Additionally, the implied total metal mass of the CGM is roughly equivalent to the metal deficiencies determined analytically for L* galaxies (Zahid et al. 2012; Peeples et al. 2014). Thus, the simple, seemingly self-consistent picture that emerges is one in which most of the cool gas observed in the halos of galaxies originates from the galaxy itself, building up over time to create a massive reservoir of 10⁴ K halo gas.

In light of the gas densities indicated by the photoionization modeling, however, this simple picture unravels. Gas densities between 10⁻⁴ and 10⁻³ cm⁻³ such as we determine are not only inconsistent with the MB04 two-phase model, but are also at least an order of magnitude lower than most cosmological simulations seem to require for the cool clouds to survive on cosmological timescales. Regardless of the feedback prescription and origin, simulations of the CGM at all redshifts typically predict that the cool, 10⁴ K gas is at least 10⁻² cm⁻³ (Stinson et al. 2012; Shen et al. 2012; Cen 2012; Hummels et al. 2013). Effectively, the problem amounts to a lack of pressure support. At densities below 10⁻²–10⁻³ cm⁻³, cool clouds do not survive longer than several million years (Joung et al. 2012). Thus, it appears that at the densities we measure, the gas would not survive on gigayear timescales, slowly assembling into a massive reservoir of cool circumgalactic gas. However, recent work by Ford et al. (2013) indicates that low ionization state metal lines primarily arise from so-called recycled outflows—gas that has been ejected, re-accreted, and ejected again from the central galaxy. The typical density of recycled material in this simulation is >10⁻² cm⁻³ within 30 kpc, but consistent with 10⁻³–10⁻⁴ beyond these innermost regions. They claim that because the gas is not in hydrostatic equilibrium, it is falling back down onto the galaxy. In this picture, much of the low-ionization state material was enriched at early times and ejected to distances far from the galaxy (into the IGM), and is now falling back down. Our results are not inconsistent with this picture.

Owing to the details of the sample selection, COS-Halos galaxies are typically fairly isolated compared to the general population at z ∼ 0.2 (Tumlinson et al. 2013; Werk et al. 2012). As a final note, we point out that group environments significantly complicate any interpretation on the origin of the CGM since absorption profiles are typically kinematically very complex (Aracil et al. 2006; Tripp 2008; Burchett et al. 2013; Stocke et al. 2014). Galaxy environment and interactions surely play some role in the observed properties of the CGM (Chen et al. 2010a; Yoon & Putman 2013), though a comprehensive study of the role of environment has not yet emerged. Finally, the absence of O vi and yet the presence of cool gas in the halos of non-star-forming galaxies (Tumlinson et al. 2011) remains a puzzle, though may be due to different origins for cool and warm-hot halo gas (Ford et al. 2013).

We examined a similar model grid that incorporates the addition of ionizing photons from the central galaxy to the Haardt–Madau UV background as our input CLOUDY spectrum. We show the spectra of these sources of ionization in Figure 13 for reference. Although the radiation from the starburst99 galaxy spectral energy distribution (SED; Leitherer et al. 1999; d = 72 kpc; SFR = 1 M_☉ yr⁻¹; both median values for the COS-Halos galaxy sample; Werk et al. 2012) dominates the HM2001 UV background, the slopes of the SEDs between 1 and 4 ryd are very similar. The extent to which the galaxy SED dominates HM2001 depends on the escape fraction of ionizing photons (assumed to be 5%), the distance from the galaxy, and the SFR of the galaxy. Since the influence of the galaxy radiation field scales as SFR/d², we explored a wide range of parameter space for this additional ionization source. The total contribution to the ionizing radiation field increases substantially with lower impact parameter and higher SFR, dominating the extragalactic UV background radiation field below ∼50 kpc for modest SFRs (SFR < 1 M_☉ yr⁻¹). The overall slopes of the HM2001 and S99 SED spectra remain approximately equivalent over the 1–4 ryd range. The only effect of including ionizing photons from the central galaxy on our results is to push the derived ionization parameter at most ∼0.5 dex higher at column densities below ∼10^16.5 cm⁻², which modestly increases the estimate of the total amount of gas (neutral + ionized) in the CGM.

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Within a galaxy virial radius, cosmic ray heating could be a significant supplemental heating and ionization source to photoionization (Wiener et al. 2013). Additionally, cosmic ray feedback theory has shown that cosmic rays may impart a significant amount of momentum to the ISM in a direction away from the galaxy, potentially driving a large-scale galactic wind (Socrates et al. 2008). At some low value of the gas volume density, heating due to the cosmic ray background (CRB; H_CRB ∝ n_H) becomes more important than photoelectric heating due to the EUVB (H_EUVB ∝ $n_{\rm H}^{2}$). Thus, there would be a corresponding minimum gas density below which the CGM gas succumbs to a CRB-driven thermal runaway to extremely high, nearly relativistic temperatures. We may assess the contribution of the CRB to the heating of CGM gas using the built-in CRB in C13, which is based on observations of H$^{+}_{3}$ in the diffuse ISM of the Milky Way (Indriolo et al. 2007). The local CRB can constitute as much as 85% of the total heating for a gas volume density of 10^−3.5 cm⁻³ and 50% of the total heating for gas volume density of 10^−2.5 cm⁻³, but these numbers are highly uncertain given the large uncertainty in the local CRB. Nonetheless, these contributions imply that at the low gas densities we derive, we may be very close to a CRB-driven thermal runaway. Thus, if the CRB in the CGM of L* galaxies is similar to the local background, then heating due to cosmic rays could have a significant impact on the results we present here.

To carry out our analysis, we used the HM2001 EUVB spectrum from galaxies and quasars for ease of comparison with previous results. Yet, Haardt & Madau (2012, hereafter HM2012) have updated their 2001 synthesis models with the addition of several new components. We show the 2012 updated HM EVUB (HM2012) as a black line in Figure 13. There are significant differences between HM2012 and HM2001. Most notably, HM2001 exhibits a lower UV flux above ∼1.5 ryd, smaller spectral breaks at 1 and 4 ryd, and a flatter soft X-ray spectrum. These differences arise primarily because of reduced H i and He ii Lyman continuum absorption from a "sawtooth modulation" by the Lyman series of H i and He ii that becomes more pronounced with increasing redshift. At low redshift, the differences between HM2001 and HM2012 are less pronounced than at high redshift.

The most important difference for our purposes is that the spectral slope of HM2001 at z ∼ 0.2 declines more steeply between 2 and 4 ryd than that of HM2012 (shown in Figure 13). Below 2 ryd, the slopes of HM2001 and HM2012 are approximately the same. Above 4 ryd, HM2001 is flatter than HM2012. Thus, compared to HM2012, HM2001 is underproducing ions like Si iii and C iii relative to the lower ionization potential ions (Mg ii and Si ii are just above 1 ryd in Figure 13). Thus, gas ionization parameters need to be higher for HM2001 to produce the observed Si iii and C iii column densities. Repeating our analysis with HM2012 generally has the effect of systematically lowering our gas ionization parameters by between 0.1 and 0.4 dex. Additionally, with our prior that gas not be super-solar, the new "preferred H i" must be raised by 0.2–0.4 dex for the cases in which this prior comes into play. If the H i is known, the preferred metallicity must be raised by 0.2–0.4 dex to be consistent with ionic column densities of higher ionization state ions.

The change in the range of log U varies on a sightline by sightline basis and depends on which ions are used for the solution. For instance, sightlines that rely on Si ii/Si iii for the determination of log U do not change significantly when re-analyzed with HM2012 because Si iii has a an ionization potential of just over 2 ryd, which is very close to where the slopes of HM2001 and HM2012 start to diverge. On the other hand, sightlines that rely on C ii/C iii to estimate log U exhibit a greater change in their determined log U values because the ionization potential of C iii falls at an energy where the slopes of HM2012 and HM2001 are most different.

In Figure 13, we show with blue and green dotted lines a range of physically plausible slopes of the EUVB above 1 ryd. We have varied the slope of the EUVB above 1 ryd with power law exponents between −0.5 (green) and −3.0 (blue) to assess an overall systematic error in our methodology. In general, a shallower slope of the EUVB tends to decrease the best-fitting ionization parameter to the measured ionic column densities of adjacent low-ions. However, the best fitting H i column density (for cases in which the AODM lower limits would give rise to a super-solar gas metallicity) also rises equivalently for many sightlines. The average systematic error in log U that arises from uncertainty in the slope of the EUVB is ±0.3 dex, on average. The effect on the baryonic mass estimate is less pronounced. For example, a the total baryonic mass estimates of the cool CGM are a factor of 2.4 lower when we do the analysis with the shallowest slope considered (green dotted line in Figure 13, power law exponent of −0.5) than the steepest slope considered (blue dotted line in Figure 13, power law exponent of −3.0). Using HM2012 lowers our baryonic mass estimates made with HM2001 by a factor of 1.25. We also examined models with non-equilibrium cooling and collisional ionization equilibrium from Gnat & Sternberg (2007), neither of which offer solutions consistent with our low-ion absorption line data.

A similar analysis incorporating CLOUDY modeling of absorption line data was carried out by Lehner et al. (2013) for 28 Lyman limit systems at z < 1, with a focus on the metallicity distribution of the gas of H i-selected sightlines. Our analysis is distinct in several important ways. First, our sample selection is based on the COS-Halos survey which is galaxy-selected (Werk et al. 2012; Tumlinson et al. 2013), and probes circumgalactic gas of L ∼ L* galaxies at distances of 10–160 kpc from the host galaxies. As a result, we probe a larger range of H i column densities (Figure 14). Second, many of our H i column density measurements are limited by line saturation, and we do not always obtain spectral coverage of the full Lyman series. The main side effect of the uncertainty in the H i column densities is a concomitant uncertainty in metallicity measurements. Thus, we limit the bulk of our analysis to a discussion of the ionization parameter of the gas, which is mostly unaffected by the uncertainty in the H i column density. Figure 15 highlights that while the solution for the ionization parameter of the gas is the same across several orders of magnitude of H i column density, the solution for the metallicity is over 1.5 dex lower for the higher H i column density gas. Given the uncertainty in the H i measurements, the metallicity of the gas is highly uncertain. The ionization parameter, however, is mostly independent of the choice of H i and metallicity.

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In this analysis, we have constrained the CLOUDY photoionization modeling with three key priors: (1) we assume solar relative abundances of the metals considered (C, N, O, Mg, Si, Fe); (2) we do not consider gas metallicities above solar; (3) we assume a log U less than −1, corresponding to a total gas density (n_h) > ∼ 10^−5.5 cm⁻³. Regarding the first prior, we acknowledge the possibility of non-solar abundance ratios by up to a factor of two (i.e., from dust depletion), which could, in practice, influence the derived metal abundances by the same factor. The ionization parameter of the gas is largely unaffected by departures from solar relative abundance ratios for two reasons. The first is that we frequently obtain column densities for two or more ions of the same element. We obtain the same ionization parameters (within the uncertainty given) when we consider only same-element pairs as we do when we consider all available ions. The second is that the CLOUDY model curves of ionic column density with log U are very steep, such that small changes in relative abundances have a minimal effect on the derived log U. Regarding the second prior, we also acknowledge the possibility of super-solar circumgalactic gas metallicities, for example, in material recently ejected by a galaxy wind, or material stripped from a nearby interacting galaxy. In general, the COS-Halos sample was selected against galaxies with any indication of a recent merger event (Tumlinson et al. 2013), though at redshifts >0.1, it can be difficult to obtain the detailed morphological measurements necessary to rule them out completely. We note that none of our systems analyzed require super-solar abundances, and many of them do not allow it given the measured ionic ratios. This constraint affects approximately one-fifth of our estimates on the ionization parameter, providing a lower bound to our estimate of log U (and thus log N_H) in these cases. In the following section, we note the specific cases for which this constraint comes into play. Regarding the third prior, the upper bound on log U of −1 rarely affects our analysis as most of the ionic ratios we consider require log U < −2. As a rule, gas with a larger (less negative) value of log U has a greater ionized gas fraction. A greater ionized gas fraction, in turn, implies a greater total mass of gas contained in the CGM of L ∼ L* galaxies. Thus, this limitation, if it has any impact on our analysis, has the effect of making our final mass estimates more conservative.

In cases of saturated H i absorption lines, we attempt to place additional constraints on the H i column density limits from the COS-Halos survey in several ways, described in detail in the sightline-by-sightline analysis, and summarized broadly here. In general, we assume the lowest value of the H i column density allowed by the COS data (Thom et al. 2012). To obtain an additional upper limit on the H i column density, we determine whether the H i absorption line profile shows the presence of visible damping wings, the production of which generally occurs at column densities greater than 10^18.5 cm⁻². In the absence of damping wings, we place this additional upper limit on H i column densities. Second, we incorporate an allowed range of Doppler b values, parameterizing the width of the absorption components to be between 1 and 70 km s⁻¹ and model the Voigt profiles of the observed H i absorption lines within the allowed range of values. This limitation is most valuable in placing additional lower limits on the H i column density, where a single saturated component cannot have a b value greater than 70 km s⁻¹. This analysis is consistent with the fits to the H i absorption lines presented in Tumlinson et al. (2013), which often give column densities above the simple AODM method determinations. Though we do not limit ourselves to the results of the Voigt profile fitting, we are guided by them. Finally, on a case by case basis, we examine the column densities of the low ionization state metal absorption lines to see if their strengths are consistent with our model grid ranges for the adopted H i value. In general, we choose a value for the adopted H i to be as low as it can be while being consistent with the line profile fits presented by Tumlinson et al. (2013) and the requirements outlined above. We show these adopted values as they compare to the COS-Halos AODM column densities in Figure 14(a). Overall, these conservative choices mean our overall mass estimate for the CGM of L* galaxies will also be conservative.

The figures shown below, beginning with Figure 16, for each system display a cross section of the CLOUDY grid at the preferred log N_H i and [X/H] determined by examining the absorption present in the COS data for each sightline and comparing it against our CLOUDY grid of output models in several different parameter spaces. These figures show the column density of the ions in the grid as a function of the most relevant quantity to our CGM mass determination, log U. On the top x-axis, we show the gas volume density, n_H = Φ/Uc. Here, Φ is the total flux of ionizing photons (∼1.21 × 10⁴ cm⁻² s⁻¹), as defined by the Haardt & Madau (2001) EUVB, and c is the speed of light. We hold both the H i column density and the metallicity constant at their preferred values (described above), given in the lower right of the figure. We incorporate our measured values for specific lower ionization state metal absorption lines into these figures by outlining in bold the allowed column density from the COS data over the grid column density curves. In cases where we have a good column density constraint on an absorption line, we place in bold the value plus or minus the error on the measurement. Upper and lower limits are indicated in bold over the full allowed range of column density. On all of these plots, we have placed a bright yellow stripe of the range of log U values that fit all the available data given the preferred log N_H i and [X/H]. In some cases, the stripe may appear to be larger than the intersection of the low ion column density measurements constrain it to be since we add additional uncertainty depending on how well we can constrain log N_H i and [X/H]. Here we assess each line of sight and its solution in the CLOUDY model grid, ultimately rating it with a quality flag between 1 and 5 (5 being the best, described in Section 1 of the main text body). We show O vi model lines and column density measurements for reference, but do not require that the selected ionization parameter account for the total observed column density.

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We describe which metal ions are most useful in constraining the ionization parameter and metallicity, and how we arrived at our adopted H i and [X/H] values, and ultimately how we constrain the ionization parameter of the circumgalactic gas. Higher values of log U imply higher ionization corrections to the H i, and higher CGM mass. For our conservative mass estimate, we therefore adopt the lowest allowed log U, the range of which is given in Section 4.2.1. When log N_H i is not well constrained by the COS-Halos data, we chose a solution for log U at the lowest log N_H i possible. The effect of this conservative choice often requires adopting the highest allowable value of the gas metallicity.