MAGiiCAT III. INTERPRETING SELF-SIMILARITY OF THE CIRCUMGALACTIC MEDIUM WITH VIRIAL MASS USING Mg ii ABSORPTION

Our sample comprises the 182 "isolated" galaxies in the "Mg ii Absorber-Galaxy Catalog" (MAGiiCAT; Nielsen et al. 2013a, Paper I). Each galaxy has a published spectroscopic redshift, with the sample spanning the range 0.07 ⩽ z ⩽ 1.12. The galaxy–quasar impact parameters range from 5.4 ⩽ D ⩽ 194 kpc. The ab absolute B- and K-band magnitudes cover the ranges −16.1 ⩾ M_B ⩾ −23.1 and −17.0 ⩾ M_K ⩾ −25.3, with rest-frame B−K colors 0.04 ⩽ B − K ⩽ 4.09. The range of detected rest-frame Mg ii λ2796 equivalent widths is 0.03 ⩽ W_r(2796) ⩽ 2.90 Å with one system at W_r(2796) = 4.42 Å. Upper limits (3σ) on W_r(2796) were measured for 59 of the 182 systems over the range W_r(2796) ⩽ 0.003 Å to W_r(2796) ⩽ 0.3 Å. Apart from the details of how the virial masses of the galaxies have been determined, which we present in this work, the particulars of the galaxy–absorber sample and standardization of photometric and absorption properties have been presented in Paper I (Nielsen et al. 2013a).

In Table 1, we present the data employed for this work. Columns 1 through 4 list the quasar field name (B1950 designation or identification of a quasar as having been discovered in the Sloan Digital Sky Survey (SDSS)), the quasar J2000 designation, the galaxy redshift, z_gal, and the impact parameter, D. Column 13 lists the Mg ii λ2796 rest-frame equivalent width, W_r(2796). These data are taken from Paper I (Nielsen et al. 2013a). The remaining columns, which are newly published data, are: (5) the galaxy r-band absolute ab magnitude, M_r, (6) the virial mass, M_h, (7) the maximum circular velocity, $V_{c}^{\rm max}$, (8) the virial radius, R_vir, (9) the ratio η_v = D/R_vir, (10) the theoretical cooling radius, R_c, (11) the ratio η_c = D/R_c, and (12) the ratio R_c/R_vir.

Table 1. Galaxy Properties

(1)	(2)	(3)	(4)	(5)	(6)	(7)a	(8)a	(9)a	(10)a,b	(11)a	(12)a	(13)
Field	J-Name	zgal	D	Mr	log M h/M☉	$V_c^{\rm max}$	Rvir	ηv	Rc	ηc	Rc/Rvir	Wr(2796)
			(kpc)	(ab)		(km s−1)	(kpc)		(kpc)			(Å)
0002 − 422	J000448.11 − 415728.8	0.8400	53.8	−21.7	$12.1_{-0.1}^{+0.2}$	$262_{-26}^{+35}$	$218_{-24}^{+32}$	$0.25_{+0.03}^{-0.03}$	$50_{+ 3}^{ -4}$	$1.07_{-0.06}^{+0.09}$	$0.23_{+0.03}^{-0.04}$	4.422 ± 0.002
0002 + 051	J000520.21 + 052411.80	0.2980	59.2	−20.9	$12.0_{-0.2}^{+0.3}$	$211_{-26}^{+45}$	$191_{-26}^{+45}$	$0.31_{+0.06}^{-0.05}$	$103_{+ 5}^{ -7}$	$0.57_{-0.02}^{+0.04}$	$0.54_{+0.08}^{-0.13}$	0.244 ± 0.003
0002 + 051	J000520.21 + 052411.80	0.5920	36.0	−22.0	$12.3_{-0.2}^{+0.2}$	$291_{-29}^{+38}$	$257_{-28}^{+37}$	$0.14_{+0.02}^{-0.02}$	$59_{+ 4}^{ -4}$	$0.61_{-0.04}^{+0.05}$	$0.23_{+0.03}^{-0.04}$	0.102 ± 0.002
0002 + 051	J000520.21 + 052411.80	0.8518	25.9	−21.2	$11.8_{-0.2}^{+0.2}$	$220_{-24}^{+40}$	$179_{-22}^{+36}$	$0.14_{+0.02}^{-0.02}$	$60_{+ 3}^{ -5}$	$0.43_{-0.02}^{+0.04}$	$0.33_{+0.05}^{-0.07}$	1.089 ± 0.008
SDSS	J003340.21 − 005525.53	0.2124	21.7	−21.3	$12.2_{-0.2}^{+0.2}$	$232_{-27}^{+41}$	$214_{-27}^{+42}$	$0.10_{+0.02}^{-0.01}$	$107_{+ 4}^{ -6}$	$0.20_{-0.01}^{+0.01}$	$0.50_{+0.07}^{-0.10}$	1.050 ± 0.030
SDSS	J003407.34 − 085452.07	0.3617	33.1	−20.1	$11.7_{-0.2}^{+0.4}$	$176_{-24}^{+55}$	$154_{-23}^{+54}$	$0.21_{+0.06}^{-0.04}$	$106_{+ 5}^{ -9}$	$0.31_{-0.01}^{+0.03}$	$0.69_{+0.12}^{-0.24}$	0.480 ± 0.050
SDSS	J003413.04 − 010026.86	0.2564	30.4	−20.7	$11.9_{-0.2}^{+0.3}$	$195_{-25}^{+47}$	$176_{-25}^{+47}$	$0.17_{+0.04}^{-0.03}$	$112_{+ 5}^{ -7}$	$0.27_{-0.01}^{+0.02}$	$0.63_{+0.10}^{-0.17}$	0.610 ± 0.060
0058 + 019	J010054.15 + 021136.52	0.6128	29.5	−19.8	$11.4_{-0.2}^{+0.4}$	$151_{-20}^{+51}$	$125_{-18}^{+47}$	$0.24_{+0.06}^{-0.04}$	$92_{+ 4}^{ -8}$	$0.32_{-0.01}^{+0.03}$	$0.74_{+0.12}^{-0.28}$	1.684 ± 0.004
0058 + 019	J010054.15 + 021136.52	0.6800	45.6	−21.2	$11.9_{-0.2}^{+0.2}$	$225_{-25}^{+42}$	$190_{-24}^{+40}$	$0.24_{+0.04}^{-0.03}$	$69_{+ 4}^{ -5}$	$0.66_{-0.03}^{+0.05}$	$0.36_{+0.05}^{-0.08}$	<0.003
SDSS	J010135.84 − 005009.08	0.2615	50.9	−21.4	$12.2_{-0.2}^{+0.2}$	$242_{-28}^{+40}$	$223_{-28}^{+40}$	$0.23_{+0.03}^{-0.03}$	$99_{+ 4}^{ -5}$	$0.51_{-0.02}^{+0.03}$	$0.44_{+0.06}^{-0.08}$	<0.110
SDSS	J010156.32 − 084401.74	0.1588	28.4	−19.2	$11.3_{-0.2}^{+0.6}$	$121_{-17}^{+64}$	$106_{-16}^{+63}$	$0.27_{+0.10}^{-0.05}$	$146_{+ 6}^{-15}$	$0.20_{-0.01}^{+0.02}$	$1.38_{+0.25}^{-0.82}$	0.360 ± 0.030
SDSS	J010352.47 + 003739.79	0.3515	48.3	−20.1	$11.7_{-0.2}^{+0.4}$	$178_{-24}^{+54}$	$157_{-23}^{+53}$	$0.31_{+0.08}^{-0.05}$	$107_{+ 5}^{ -9}$	$0.45_{-0.02}^{+0.04}$	$0.68_{+0.12}^{-0.23}$	0.380 ± 0.030
0102 − 190	J010516.82 − 184641.9	1.0250	40.0	−22.3	$12.1_{-0.1}^{+0.1}$	$284_{-25}^{+31}$	$230_{-22}^{+27}$	$0.17_{+0.02}^{-0.02}$	$36_{+ 3}^{ -3}$	$1.12_{-0.08}^{+0.11}$	$0.16_{+0.02}^{-0.02}$	0.670 ± 0.050
0109 + 200	J011210.18 + 202021.79	0.5340	44.7	−20.4	$11.6_{-0.2}^{+0.4}$	$173_{-23}^{+53}$	$147_{-21}^{+50}$	$0.30_{+0.08}^{-0.05}$	$92_{+ 4}^{ -8}$	$0.49_{-0.02}^{+0.05}$	$0.63_{+0.11}^{-0.22}$	2.260 ± 0.050

Notes. ^aUncertainties are based upon uncertainties in the virial masses (Column 6). For some quantities a larger (smaller) virial mass results in smaller (larger) value such that the uncertainties anti-correlate. ^bBecause the slope of R_c changes sign as function of virial mass, where the slope is positive the uncertainties correlate and where the slope is negative they anti-correlate (see Figure 11). In the narrow virial mass ranges where the slope of R_c changes sign, it is possible that both the upward and downward uncertainties in virial mass can result in an upward (or downward) uncertainty in R_c.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The M_r were computed using the methods applied to obtain M_B and M_K as described in Paper I (Nielsen et al. 2013a). The resulting range is −22.2 ⩽ M_r ⩽ −16.4. Calculation of the virial radius, R_vir, was presented in Churchill et al. (2013). Calculation of the theoretical cooling radius is discussed in Section 3.4.

Here, we elaborate on the method employed to determine the galaxy virial masses that were originally studied in Churchill et al. (2013). For each galaxy in the sample, the virial mass (dark + baryonic matter), M_h, was obtained by halo abundance matching. The virial mass is the total mass enclosed within the virial radius. The virial radius is defined as the radius enclosing an average density Δ_c(z)ρ_c, where Δ_c(z) (see Equation (A15) of Eke et al. 1996) is a cosmology and redshift dependent multiplier under the assumption of virialization of a collapsed spherical top-hat perturbation, and ρ_c is the critical density.

Halo abundance matching assigns galaxies to dark matter halos in a simulation based on number density with no free parameters. The method has been thoroughly explored and applied to various astronomical problems (Kravtsov et al. 2004; Tasitsiomi et al. 2004; Vale & Ostriker 2004; Conroy et al. 2006; Conroy & Wechsler 2009; Guo et al. 2010; Behroozi et al. 2010; Firmani et al. 2010; Trujillo-Gomez et al. 2011; Rodriguez-Puebla et al. 2012; Behroozi et al. 2013; Moster et al. 2013; Reddick et al. 2013). In practice, the technique has been extremely successful in reproducing many galaxy statistics, such as the two-point correlation function as a function of redshift (Conroy et al. 2006), luminosity (Trujillo-Gomez et al. 2011), stellar mass (Reddick et al. 2013), and color (Hearin & Watson 2013, accounting for halo formation times), as well as the luminosity-velocity relation, baryonic Tully–Fisher relation, and galaxy velocity function (Trujillo-Gomez et al. 2011). Halo abundance matching also yields galaxy stellar-to-halo mass relations that agree with direct estimates from lensing and satellite kinematics within the uncertainties of the observations (see Dutton et al. 2010).

In essence, halo abundance matching links a given property (i.e., stellar mass, luminosity, etc.) of galaxies to a given halo property (circular velocity, virial mass, etc.) in a monotonic fashion. For this work, the dark matter halo catalogs are taken from the Bolshoi N-body cosmological simulation (Klypin et al. 2011).

For the halo property, we adopt the maximum circular velocity,

$$\begin{equation} V_{\rm c}^{\rm max} = \left[ \frac{{\it GM}_{\rm \, h}(\!<\!r)}{r} \right] ^{1/2} \, \left |\right. _{\rm max} , \end{equation} \tag{ 1 }$$

which properly accounts for the depth of the galactic potential and is unambiguously defined for both central halos and sub-halos (halos within the virial radius of larger halos, Trujillo-Gomez et al. 2011). At a given redshift, the halo catalog comprises individual halos for which both $V_c^{\rm max}$ and M_h are tabulated.

For the galaxy property, we adopt the r-band luminosity, M_r. For the number density of galaxies with a given M_r, we adopt the COMBO-17 r-band luminosity function (LF) of Wolf et al. (2003), which covers the redshift of our galaxy sample in a band that successfully reproduces the clustering of galaxies at both low and high redshifts (Trujillo-Gomez et al. 2011; Gerke et al. 2013).

For the galaxy sample, we solve for the $V_c^{\rm max}$ for a galaxy with M_r such that the fractional area under the observed galaxy LF corresponds to an equal fractional area under the curve of the distribution of maximum circular velocities of halos,

$$\begin{equation} \frac{1}{N_{M_r}(z\,)} \int _{-\infty }^{M_r} \!\!\!\!\!\!\! n(M_{r},z\,) \, dM_r = \frac{1}{N_{V_c^{\rm max}}(z\,)} \int _{V_c^{\rm max}}^{\infty } \!\!\!\!\!\!\!\! n(V_c^{\rm max}\!,z\,) \, dV_c^{\rm max} , \end{equation} \tag{ 2 }$$

where the denominators are the total number density in the respective distributions. The LF is preserved by construction. The only assumption in the method is that there is only one galaxy inhabiting each dark matter halo.

The redshift of a given galaxy determines the redshift of both the Bolshoi halo catalog and the LF for which Equation (2) was applied. Wolf et al. (2003) published the r-band LFs for five redshifts, z = 0.3, 0.5, 0.7, 0.9, and 1.1. We abundance match a given galaxy M_r to $V_c^{\rm max}$ in a Δz = 0.2 redshift bin bracketing the galaxy redshift, where the bin centers correspond to the five COMBO-17 redshifts. For z < 0.2, we opted to not use the "local" r-band LFs from SDSS (Blanton et al. 2001) or 2dFGRS (Madgwick et al. 2002) due to inconsistencies with the COMBO-17 LF, which may be due to different sensitivities of the surveys at the bright end (see Wolf et al. 2003). To maintain self-consistency, we adopt the COMBO-17 LF in the bin 0.2 < z < 0.4 under the assumption that the LF does not evolve significantly below z = 0.3.

There is intrinsic scatter in $V_c^{\rm max}$ for a given M_h due to variation in formation times of halos of the same mass (see Trujillo-Gomez et al. 2011). Once the halo abundance matching is solved (a $V_c^{\rm max}$ for each dark matter halo in the Bolshoi catalog at the appropriate redshift is assigned to an M_r for a galaxy in the sample), we account for the scatter in M_h with $V_c^{\rm max}$ by computing the average M_h of all the halos that fall in a fixed luminosity bin, ΔM_r, centered on the measured value for that galaxy. We adopted ΔM_r = 0.1 (for details see Appendix A). Since halo abundance matching is a statistical method, each derived M_h should be interpreted as the average mass of a halo which hosts a galaxy of a given M_r.

In Columns 6–8 of Table 1, we present the galaxy properties derived from halo abundance matching. The resulting virial masses have the range 10.7 ⩽ log M_h/M_☉ ⩽ 13.9. Including both systematics and scatter, the uncertainties are δlog M_h ≃ 0.1 at log M_h/M_☉ = 10 increasing quasi-linearly to δlog M_h ≃ 0.35 at log M_h/M_☉ = 13. However, for each galaxy, we adopt the 1σ standard deviation in the scatter of the average M_h in the luminosity bin as the uncertainty in M_h.

We obtained the virial radius, R_vir, for each galaxy using the relation with M_h given by Bryan & Norman (1998). The resulting virial radii have the range 70 ⩽ R_vir ⩽ 800 proper kpc. The uncertainties in R_vir were obtained from the uncertainties in the virial masses using standard error propagation. The typical uncertainty is δR_vir/R_vir ≃ 0.1.

In Appendix A, we quantify and discuss the systematic and statistical uncertainties in M_h associated with our methodology and quantify the effects of observational uncertainties.

For Mg ii absorption, many works have measured the luminosity dependence of the "absorption radius" assuming the Holmberg scaling R(L) = R_*(L/L*)^β, where R_* is the absorption radius of an L* galaxy and β parameterizes the luminosity scaling (e.g., Nielsen et al. 2013b and references therein). The "absorption radius" is interpreted as an average physical extent out to which absorption is detected above a given absorption threshold; it represents an idealistic projected radius within which CGM gas is detected and outside of which CGM gas is not detected.

In Paper II (Nielsen et al. 2013b), the two parameters R_* and β were obtained for various absorption thresholds, W_cut, by maximizing the number of systems with W_r(2796) ⩾ W_cut residing at D ⩽ R(L) and maximizing the number of systems with W_r(2796) < W_cut residing at D > R(L). The covering fraction, f_c, of the absorption within R(L) for each threshold is also directly computed in their analysis, where the uncertainties are determined using binomial statistics (see Gehrels 1986).

Following the methods applied by Nielsen et al. (2013b), we investigated whether there is a virial mass dependence of the Mg ii CGM absorption radius, R(M_h), for the four W_r(2796) absorption thresholds, W_cut = 0.1, 0.3, 0.6, and 1.0 Å. In place of the galaxy luminosity relative to L*, we define $M_{\rm \, h}^{\ast } = 10^{12}$ M_☉ (the median mass of the sample) and write

$$\begin{equation} R(M_{\rm \, h}) = R_{\ast } \left(\frac{M_{\rm \, h}}{M_{\rm \, h}^{\ast }} \right) ^{\gamma } . \end{equation} \tag{ 3 }$$

In Figure 1, we plot D versus $M_{\rm \, h}/M_{\rm \, h}^{\ast }$ for the sample. In each panel, purple points have W_r(2796) ⩾ W_cut and green points have W_r(2796) < W_cut. Thus, purple points indicate absorption detections and green points indicate absorption "misses" for the given W_cut. Open points indicate that the measurement of W_r(2796) is an upper limit to the detection sensitivity of the quasar spectrum. The solid line is the maximum likelihood fit to Equation (3) and shows the virial mass dependence of the Mg ii CGM "absorption radius," R(M_h). The dashed curves provide the 1σ envelope to the best fit parameters. Note that there are fewer data points included in the analysis for the W_cut = 0.1 Å subsample. This is because we exclude non-detections with W_r(2796) limits greater than W_cut. The fitting results suggest that the absorption radius of an $M_{\rm \, h}^{\ast }$ galaxy, R_*, decreases with increasing W_cut, from ≃ 70 kpc for W_cut = 0.1 Å to ≃ 50 kpc for W_cut = 1.0 Å. However, the uncertainties in R_* are larger, which reflects the degree the absorption radius is actually a "fuzzy" boundary. As the equivalent width threshold, W_cut, is raised, we find that the virial mass dependence systematically decreases from γ ≃ 0.45 for W_cut = 0.1 Å to γ ≃ 0.20 for W_cut = 1.0 Å.

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The resulting R_* and γ values can be applied to quantify the absorption radius relative to the virial radius. We will refer to this quantity as the "mass-normalized absorption envelope," and denote the quantity as η_v(M_h). Defining η_v = D/R_vir and $\eta ^{\ast }_{\rm v} = R_{\ast }/R^{\, \ast }_{\rm vir}$, where $R^{\, \ast }_{\rm vir}$ is the virial radius for an $M_{\rm \, h}^{\ast } =10^{12}$ M_☉ halo taken at the median redshift of the sample, and invoking $R_{\rm vir} \propto M_{\rm \, h}^{1/3}$ (Bryan & Norman 1998), we obtain the relation

$$\begin{equation} \eta _{\rm v} (M_{\rm \, h}) = \eta ^{\ast }_{\rm v} \left(\frac{M_{\rm \, h}}{M_{\rm \, h}^{\ast }} \right) ^{\gamma ^{\prime }} , \end{equation} \tag{ 4 }$$

where γ' = γ − 1/3.

In Figure 2, we plot η_v(M_h) versus $M_{\rm \, h}/M_{\rm \, h}^{\ast }$ for the W_r(2796) absorption thresholds, W_cut = 0.1, 0.3, 0.6, and 1.0 Å. The mean mass-normalized absorption envelope for $M_{\rm \, h}^{\ast }$ galaxies is $\eta ^{\ast }_{\rm v} \simeq 0.3$ and is remarkably consistent within uncertainties as being independent of the absorption threshold. However, the mean covering fraction decreases by a factor of two as W_cut is increased from 0.1 to 1.0 Å. The virial mass dependence is quite weak, ranging from γ' ≃ +0.1 to ≃ − 0.14 as W_cut is increased. Overall, by scaling the absorption radius parameters, we find that the parameters describing the mass-weighted absorption envelope, $\eta ^{\ast }_{\rm v}$ and γ', indicate a very weak dependence on virial mass and that this holds for all absorption thresholds.

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In view of the fitted relation log W_r(2796)∝ − 2log (D/R_vir) obtained by Churchill et al. (2013), and given that log R_vir∝(1/3)log M_h, one could infer that log W_r(2796)∝ − 2log D + (2/3)log M_h. That is, log W_r(2796)∝ − 2log D for a narrow range of M_h and log W_r(2796)∝(2/3)log M_h for a narrow range of D. This behavior is consistent with the virial mass segregation on the W_r(2796)–D plane presented by Churchill et al. (2013), in which stronger absorption is preferentially associated with higher mass halos and found at larger impact parameter.

To further investigate the relationships between W_r(2796), virial mass, and impact parameter, we explored the W_r(2796)–D plane for differential virial mass behavior, and the W_r(2796)–M_h plane for differential impact parameter behavior. In Figure 3(a), we present the W_r(2796)–D plane in which we colored the data points according virial mass using the mass decades log M_h/M_☉ = 10–11, 11–12, 12–13, and 13–14. Consistent with many works (see Nielsen et al. 2013a and references therein), W_r(2796) tends to decrease with increasing impact parameter. There is a clear visual trend for higher mass halos to host larger W_r(2796). This is especially apparent in that the "upper envelope" of W_r(2796) is dominated by the higher mass galaxies, i.e., log M_h/M_☉ > 12. Furthermore, it appears that the slope of each upper envelope increases with increasing virial mass.

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To quantify this differential virial mass behavior in the upper absorption envelope, we used a maximum likelihood approach to solving the relation W_r(2796) = α₁log D + α₂, for each of the four virial mass ranges presented in Figure 3(a). We minimized the function

$$\begin{equation} {\cal L}(\alpha _1,\alpha _2) = {\rm min} \left| \, N_{\rm abv}/N_{\rm tot} - {\rm erfc}(1) \, \right| , \end{equation} \tag{ 5 }$$

where N_abv/N_tot is the ratio of systems above the envelope to the total number of systems in the mass range. The complimentary error function is employed to account for the scatter in the data and the different number of data points in each virial mass range. We allow 15.7% (1σ) of the data points in each mass range to reside above the envelope when ${\cal L}(\alpha _1,\alpha _2)$ is a minimum. Thus, the resulting envelope encloses 84.3% of the data. The envelopes have all been normalized to W_r(2796) = 0 Å at D = 200 kpc.

The resulting fitted parameters are listed in Table 2, as are the values of the likelihood function. The envelopes for each of the respective virial mass ranges are plotted on Figure 3(a). The exercise quantifies the degree to which the upper absorption envelope on the W_r(2796)–D plane is virial mass dependent. At a given impact parameter, larger W_r(2796) tends to arise in higher mass halos.

Table 2. Envelope W_r(2796) = α₁log D + α₂

(1)	(2)	(3)	(4)
log M h/M☉	α1	α2	${\cal L}(\alpha _1,\alpha _2)$
(10–11)	$-0.7^{+0}_{-0.2}$	$1.6^{+0.2}_{-0.5}$	1.57 × 10−1
(11–12)	$-1.3^{+0.1}_{-0.2}$	$3.0^{+0.3}_{-0.6}$	2.54 × 10−3
(12–13)	$-2.1^{+0.1}_{-0.3}$	$4.8^{+0.1}_{-0.2}$	3.62 × 10−3
(13–14)	$-3.7^{+0.2}_{-0.1}$	$8.5^{+0.2}_{-0.4}$	3.23 × 10−2

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In Figure 3(b), we show the W_r(2796)–M_h plane where we have color coded the data points binned by impact parameter using the bins 0 < D ⩽ 25 kpc, 25 < D ⩽ 50 kpc, 50 < D ⩽ 100 kpc, and 100 < D ⩽ 200 kpc. Again, for a fixed impact parameter range, there is a clear general trend for increasing W_r(2796) with increasing virial mass. Though a range of W_r(2796) are present at a given virial mass in each impact parameter range, the upper envelope of the absorption for an impact parameter range clearly increases with virial mass. Parameterizing the envelope as W_r(2796) = α₁log M_h/M_☉ + α₂ for each impact parameter range, we applied Equation (5) to quantify this behavior.

The resulting fitted parameters and values of the likelihood function are listed in Table 3. The results indicate that, in a finite range of impact parameter, stronger W_r(2796) absorption is preferentially found around higher mass halos and that this trend is more pronounced for smaller impact parameters (i.e., D < 50 kpc). Though the data exhibit substantial scatter, the trends highlighted in Figures 3(a) and (b) are consistent with the notion of a correlation between virial mass and Mg ii equivalent width at a fixed impact parameter. In order to further investigate the relationships between W_r(2796), virial mass, and impact parameter, we examined the behavior of the means in these quantities.

Table 3. Envelope W_r(2796) = α₁log (M_h/M_☉) + α₂

(1)	(2)	(3)	(4)
D, kpc	α1	α2	${\cal L}(\alpha _1,\alpha _2)$
(0–25]	$1.0^{+0.1}_{-0.1}$	$-10.0^{+1.5}_{-1.9}$	3.45 × 10−3
(25–50]	$0.8^{+0.1}_{-0.1}$	$-8.0^{+2.2}_{-2.1}$	4.47 × 10−3
(50–100]	$0.5^{+0.2}_{-0.1}$	$-5.3^{+1.1}_{-2.1}$	6.80 × 10−2
(100–200)	$0.3^{+0.2}_{-0.1}$	$-3.3^{+1.1}_{-2.2}$	1.04 × 10−3

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In Figure 4(a), we present the W_r(2796)–D plane in which we plot the mean W_r(2796) in a fixed impact parameter range for the virial mass decades log M_h/M_☉ = 10–11, 11–12, 12–13, and 13–14. The impact parameter ranges are 0 < D ⩽ 25 kpc, 25 < D ⩽ 50 kpc, 50 < D ⩽ 100 kpc, and 100 < D ⩽ 200 kpc. Vertical bars provide the 1σ variances in the mean W_r(2796) and horizontal bars indicate the width of the impact parameter bin. Note that in the virial mass decade log M_h/M_☉ = 10–11 (blue points), there are only three galaxies in the sample.

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The general trend seen in Figure 4(a) is that, in each impact parameter bin, the mean W_r(2796) increases with virial mass. We further illustrate the trend in the four panels of Figure 4(b), where we plot the mean W_r(2796) as a function of virial mass in fixed impact parameter bins. The points are plotted at the mean virial mass and mean W_r(2796). The data clearly show a trend of increasing mean W_r(2796) as a function of M_h in each impact parameter bin.

Since the appearance of binned data can be sensitive to the choice of binning, and since we clearly do not have equal numbers of galaxies in each virial mass decade, we divided the sample into virial mass tertiles and virial mass quartiles. We obtain the same qualitative results presented in Figure 4.

To determine whether the correlations between W_r(2796) and M_h in each impact parameter range are statistically significant, we performed a non-parametric rank correlation test on the unbinned data represented in each panel of Figure 4(b). Since a substantial number of our W_r(2796) values are upper limits, we employed the Brown, Hollander, & Korwar bhk-τ test (Brown et al. 1974), which allows for upper limits in either the dependent or the independent variable (also see Feigelson & Nelson 1985; Isobe et al. 1986; Wang & Wells 2000). The tests do not significantly rule out the null hypothesis of no correlation between W_r(2796) and M_h to better than 3σ. However, they suggest a strong trend to better than 2.5σ in each impact parameter range.

In view of the prediction that log W_r(2796) is proportional to (2/3)log M_h − 2log D, we might expect the data presented in Figure 4(b) would obey log W_r(2796) = (2/3)log M_h + C in finite impact parameter bins. Assuming the relation log W_r(2796) = α₁log M_h/M_☉ + α₂, we applied a maximum likelihood linear fit to the unbinned data presented in each panel of Figure 4(b) to obtain an estimate of the slope between W_r(2796) and M_h in fixed impact parameter bins. Accounting for upper limits for some of our W_r(2796) measurements, we employed the Expectation-Maximization algorithm emalgo (Wolynetz 1979).

The resulting fitted parameters are presented in Table 4. Columns 4–7 provide the number of galaxy–absorber pairs in each mass decade. The fits are overplotted on the binned data presented in Figure 4(b). For all impact parameter bins, we find that the slope, α₁, is always less than the predicted 2/3. However, the slopes for the D > 50 kpc bins are consistent with 2/3 within uncertainties. The best fit slope increases as impact parameter is increased, though the large uncertainties for D > 50 kpc reflect the increased scatter in W_r(2796) due to the decreasing covering fraction as impact parameter increases (Nielsen et al. 2013b). Within uncertainties, the zero-point of the fit, α₂, decreases with increasing impact parameters consistent with the D⁻² scaling.

In view of the results of Churchill et al. (2013) in which the Mg ii λ2796 equivalent width is tightly anti-correlated with η_v = D/R_vir, the projected impact parameter in units of the virial radius, we further investigate the relationship between W_r(2796), virial mass, and η_v.

Of interest is that on the W_r(2796)–D plane, there is a virial mass segregation (at the 4σ significance level; Churchill et al. 2013), in which higher virial mass galaxies are seen to have larger W_r(2796) and D than galaxies with lower virial masses. This behavior induces a substantial scatter on the W_r(2796)–D plane, and shows that the scatter is systematic with virial mass (this systematic scatter is also seen at the 4σ significance level with B- and K-luminosity, Nielsen et al. 2013b). However, when D is normalized to R_vir, the virial mass segregation vanishes on the W_r(2796)–η_v plane and the scatter is reduced to a very high significance level relative to the scatter on the W_r(2796)–D plane.

Overall, this behavior might suggest that the role of virial mass is manifest in the virial radius, such that W_r(2796) should show little to no trend with virial mass when examined as a function of η_v. In Figure 5(a), we present the W_r(2796)–η_v plane in which we plot the mean W_r(2796) for η_v ⩽ 0.3 and η_v > 0.3. The cut η_v = 0.3 is motivated by the above result (see Figure 2) in which $\eta ^{\ast }_{\rm v} \simeq 0.3$ and that virial mass scaling of η_v(M_h) is very weak. Data points are colored as in Figure 4. The mean W_r(2796) is clearly independent of virial mass for η_v ⩽ 0.3. The data do not present as clear a picture between mean absorption strength and virial mass for η_v > 0.3; however, for log M_h/M_☉ > 11, the W_r(2796) are consistent with being independent of virial mass within the 1σ variances of their distributions (note that there is only a single data point for log M_h/M_☉ < 11).

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In Figure 5(b), we plot the mean W_r(2796) directly as a function of virial mass. For both η_v ⩽ 0.3 and η_v > 0.3, bhk-τ non-parametric rank correlation tests on the unbinned data represented in each panel of Figure 5(b) are consistent with the null-hypothesis of no correlation between W_r(2796) and M_h. This is a remarkable behavior that strongly suggests a self-similarity between the cool/warm CGM over a wide range of virial mass. Within the inner third of the virial radius, the strength of the absorption is invariant with virial mass. The degree of invariance we find outside the inner third of the virial radius is also remarkable. Whatever the physical source governing the column density and kinematic distribution of Mg ii absorbing gas (chemical enrichment, stellar feedback, infall accretion, cooling and heating, and/or destruction and creation mechanisms), the net result is one in which a uniform behavior in the average properties of the gas is constant as a function of η_v for all virial masses.

In the first modern models of galaxy formation in the dark matter paradigm, theorists proposed that the cooling of gas in the halo is a key mechanism governing galaxy mass (cf. White & Rees 1978; White & Frenk 1991) and the extent and mass of cool/warm CGM gas (cf. Mo & Miralda-Escude 1996). In such models, it was stipulated that gas falling into dark matter halos shock heats and sets up an initial hot phase (T ⩾ 10⁶ K) at the virial temperature with gas density decreasing with increasing radius. The models were developed based on the notion of a theoretical cooling radius, R_c, inside of which gas cools, falls into the galaxy, and feeds star formation, and outside of which the gas does not have time to cool and remains in the hot phase. Later works show the cooling time scale in lower mass halos is shorter than the infall dynamical and/or compression time scale, and cold-mode accretion feeds the central galaxy (e.g., Birnboim & Dekel 2003; Kereš et al. 2005, 2009; Dekel & Birnboim 2006; Stewart et al. 2011; van de Voort et al. 2011).

In the model of Mo & Miralda-Escude (1996), the cool/warm gas (10⁴ ⩽ T ⩽ 10⁵ K) traced by H i and Mg ii absorption is predicted to have a high covering fraction inside the cooling radius. Maller & Bullock (2004) expanded the Mo & Miralda-Escude (1996) model to account for a multi-phase gas medium in which they allow for a hot gas core to persist at r < R_c, while invoking thermal and dynamical instabilities to provide for the fragmentation and condensation of some of the hot gas into cool clouds. This multi-phase model predicts a non-unity covering fraction of cool/warm absorbing gas inside the cooling radius, which is more in line with Mg ii absorption observations (Kacprzak et al. 2008; Chen et al. 2010; Nielsen et al. 2013b). The model also predicts that cool/warm gas which originated via condensation from the initial hot halo gas will reside exclusively inside R_c.

To investigate where the Mg ii absorbing CGM resides in relation to the theoretical cooling radius and to determine the covering fraction both inside and outside the cooling radius, we estimated R_c using the model of Maller & Bullock (2004). Other analytical dark matter halo models that predict Mg ii absorption have been developed (e.g., Tinker & Chen 2008; Chelouche et al. 2008; Chelouche & Bowen 2010), but the model of Maller & Bullock (2004) is best suited for our study because it is based upon physical principles that provide a clear formalism for computing the theoretical cooling radius as a function of virial mass.

Formally, the cooling radius is defined at the radial distance, r, from the center of the halo at which the initial gas density, ρ_gas(r), equals the characteristic density at which gas can cool, ρ_c, known as the "cooling density." As such, cooling of the gas can occur for r ⩽ R_c when ρ_gas(r) ⩾ ρ_c. The theoretical cooling radius is defined when

$$\begin{equation} \rho _{\rm gas}(R_{\rm c}) = \rho _{\rm c} \end{equation} \tag{ 6 }$$

is satisfied. Following Maller & Bullock (2004), we applied their Equation (9) for ρ_gas(r) and Equation (12) for ρ_c to obtain R_c for each galaxy in our sample. The required input quantities are the virial mass, virial radius, redshift, formation time of the dark matter halo, and metallicity of the hot gas halo. In Appendix B, we describe our computation of R_c and provide a brief discussion of how the value of R_c responds to the input quantities (see Figure 11). For our work, the most uncertain quantities are the halo formation time and the metallicity of the hot halo gas, Z_gas.

For fixed redshift, the formation time, τ_f, is shorter for higher mass halos. For fixed virial mass, τ_f decreases with increasing redshift. Since the cooling density scales as $\rho _{\rm c} \propto \tau ^{-1}_{\rm f}$, a shorter formation time yields a smaller cooling radius. We describe our estimation of the formation time in Appendix B and present τ_f as a function of redshift and virial mass in Figure 11(a).

The cooling radius is inversely proportional to the cooling function, Λ(T, Z_gas). A fixed volume of solar metallicity gas can cool at a rate 3–10 times more rapidly than zero metallicity gas, depending upon the temperature regime. Since the cooling density follows ρ_c∝Λ⁻¹(T, Z_gas), the value of R_c can be as much as a factor of ≃ 1.5 larger in a halo of the same mass and redshift but with ≃ 1 dex higher metallicity (see Figure 11(b)). However, this applies only in the lower mass halos; at higher mass, and therefore higher initial gas temperatures, the cooling rate is metallicity independent (where bremsstrahlung cooling dominates).

It is important to keep in mind that, in the framework of the Maller & Bullock (2004) model, the gas metallicity corresponds to the hot phase of the halo, for which evidence is mounting that the mixing between stellar feedback and accretion of the IGM converges on a mean metallicity of Z_gas ∼ 0.1 (cf. Crain et al. 2013). For our presentation of R_c, we adopt Z_gas ∼ 0.1, as motivate in Appendix B.

The computed theoretical cooling radii for the galaxies in our sample are listed in Column 10 of Table 1 and plotted in Figure 6(a) as a function of virial mass. Points are colored by Mg ii λ2796 equivalent width bins. Filled points are detections and open points are upper limits. In Figure 6(b), we plot the ratio R_c/R_vir for the galaxies (also see Column 12 of Table 1). Most of the galaxies in the sample have 0.1 ⩽ R_c/R_vir ⩽ 1.0, in that the cooling radius lies inside the virial radius. For log M_h/M_☉ < 11.3, the cooling radius resides outside the virial radius. The scatter in these two diagrams is due solely to the different galaxy redshifts at a given virial mass. Higher redshift galaxies have shorter formation times, and a shorter formation time yields a larger cooling density, and therefore a smaller cooling radius.

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In Figure 6(c), we plot the fractional projected distance within the theoretical cooling radius at which absorption is probed, η_c = D/R_c, versus the fractional projected distance within the virial radius, η_v = D/R_vir, at which absorption is probed. These quantities are listed in Columns 9 and 11 of Table 1. Whereas virtually all the Mg ii absorption is found within the virial radius (noting that our sample probes η_vir ⩽ 1 in all but three cases), we find that Mg ii absorption is detected well outside the theoretical cooling radius. In Figure 6(d), we present W_r(2796) versus η_c. Of interest is that the strongest absorbers are detected over a wide range of η_c, including well outside the theoretical cooling radius.

In Figure 7 and Table 5, we present the covering fraction, f_c(η_c), for different W_r(2796) absorption thresholds, W_cut, for η_c ⩽ 1.0 (inside the cooling radius) and η_c > 1.0 (outside the cooling radius) for the fiducial model with Z_gas = 0.1. For this exercise, we examined the full range of virial masses (black points), and the two subsamples defined by log M_h/M_☉ ⩾ 12, and by log M_h/M_☉ < 12, where log M_h/M_☉ = 12 is the median of the sample. We also computed f_c(η_c) for the ranges 0 ⩽ η_c ⩽ 0.5 and 0.5 < η_c ⩽ 1.

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Table 5. Covering Fraction with Cooling Radius^a

(1)	(2)	(3)	(4)	(5)
Wcut	ηc Range	fc(ηc)	fc(ηc)	fc(ηc)
(Å)	D/Rc	(All)	(M h < 1012)	(M h ⩾ 1012)
0.1	⩽0.5	$0.91_{- 0.04}^{+ 0.03}$	$0.89_{- 0.06}^{+ 0.04}$	$0.96_{- 0.09}^{+ 0.03}$
	0.5–1.0	$0.59_{- 0.09}^{+ 0.08}$	$0.32_{- 0.12}^{+ 0.14}$	$0.80_{- 0.11}^{+ 0.08}$
	⩽1.0	$0.80_{- 0.04}^{+ 0.04}$	$0.74_{- 0.06}^{+ 0.05}$	$0.88_{- 0.07}^{+ 0.05}$
	>1.0	$0.53_{- 0.10}^{+ 0.10}$	$0.29_{- 0.18}^{+ 0.26}$	$0.59_{- 0.11}^{+ 0.10}$
0.3	⩽0.5	$0.83_{- 0.05}^{+ 0.04}$	$0.83_{- 0.06}^{+ 0.05}$	$0.81_{- 0.11}^{+ 0.08}$
	0.5–1.0	$0.38_{- 0.07}^{+ 0.08}$	$0.10_{- 0.06}^{+ 0.12}$	$0.57_{- 0.11}^{+ 0.10}$
	⩽1.0	$0.66_{- 0.05}^{+ 0.04}$	$0.65_{- 0.06}^{+ 0.06}$	$0.69_{- 0.07}^{+ 0.07}$
	>1.0	$0.42_{- 0.09}^{+ 0.09}$	$0.29_{- 0.18}^{+ 0.26}$	$0.46_{- 0.10}^{+ 0.10}$
0.6	⩽0.5	$0.58_{- 0.06}^{+ 0.06}$	$0.58_{- 0.07}^{+ 0.07}$	$0.59_{- 0.12}^{+ 0.11}$
	0.5–1.0	$0.16_{- 0.05}^{+ 0.07}$	$0.00_{- 0.00}^{+ 0.09}$	$0.27_{- 0.09}^{+ 0.10}$
	⩽1.0	$0.43_{- 0.05}^{+ 0.05}$	$0.43_{- 0.06}^{+ 0.06}$	$0.42_{- 0.07}^{+ 0.08}$
	>1.0	$0.26_{- 0.07}^{+ 0.08}$	$0.25_{- 0.16}^{+ 0.24}$	$0.26_{- 0.08}^{+ 0.09}$
1.0	⩽0.5	$0.31_{- 0.05}^{+ 0.06}$	$0.24_{- 0.06}^{+ 0.07}$	$0.48_{- 0.11}^{+ 0.11}$
	0.5–1.0	$0.08_{- 0.04}^{+ 0.06}$	$0.00_{- 0.00}^{+ 0.09}$	$0.13_{- 0.06}^{+ 0.09}$
	⩽1.0	$0.23_{- 0.04}^{+ 0.04}$	$0.18_{- 0.04}^{+ 0.05}$	$0.30_{- 0.07}^{+ 0.07}$
	>1.0	$0.20_{- 0.06}^{+ 0.08}$	$0.25_{- 0.16}^{+ 0.24}$	$0.18_{- 0.07}^{+ 0.09}$

Note. ^aValues apply for Z_gas = 0.1.

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As documented in Table 5, for η_c ⩽ 0.5, the covering fraction decreases from ≃ 0.9 to ≃ 0.3 as W_cut is increased from 0.1 to 1.0 Å and shows little to no dependence on virial mass for all W_cut, except for a suggestion that higher mass halos have larger f_c(η_c) for W_cut = 1.0 Å (though the values are consistent within uncertainties). For 0.5 < η_c ⩽ 1.0, the covering fraction also decreases as W_cut is increased, but in this regime f_c(η_c) exhibits virial mass dependence such that higher mass halos have substantially higher covering fraction than lower mass halos. In fact, for W_cut > 0.6 Å, f_c(η_c) is consistent with zero in lower virial mass galaxies.

Given that R_c is strongly anti-correlated with M_h, a fixed D would probe further out into the theoretical cooling radius for higher mass halos. Thus, at fixed η_c, higher mass halos are probed at relatively smaller D than are lower mass halos. Churchill et al. (2013) showed (see their Figure 2) that the covering fraction in fixed impact parameter bins, f_c(D), was higher for higher mass halos than for lower mass halos, particularly for D > 50 kpc and W_cut = 0.1 and 0.3 Å. For D ⩽ 50 kpc, f_c(D) is effectively independent of virial mass for W_cut = 0.1, 0.3, and 0.6 Å, but is higher for higher mass halos for W_cut = 1.0 Å. This behavior resembles the behavior of f_c(η_c). Given these considerations, the virial mass dependence of f_c(η_c) in the range 0.5 < η_c ⩽ 1.0 is likely reflecting the virial mass dependence of f_c(D) on impact parameter.

As illustrated in Figure 7, the average covering fraction inside the theoretical cooling radius (η_c ⩽ 1), exhibits little to no dependence on virial mass. As the W_r(2796) absorption threshold is increased, the covering fraction decreases from ≃ 0.8 down to ≃ 0.2. The average covering fraction outside the theoretical cooling radius does not vanish, as would be expected if the absorbing gas originated from cloud fragmentation and condensation from the hot coronal halo gas. At projected distances where the density of the hot coronal gas is too low to cool, f_c(η_c) ranges from ≃ 0.5 down to ≃ 0.2, decreasing as W_cut is increased from 0.1 to 1.0 Å and showing little evidence for a virial mass dependence, especially for W_cut = 0.6 and 1.0 Å.

Comparing f_c(η_c) inside and outside the theoretical cooling radius, we can infer that the spatial properties of the cool/warm CGM gas are not fundamentally connected to where the cool/warm gas resides relative to the cooling radius of the hot coronal halo gas.⁵ The 20–50% covering fraction outside the cooling radius indicates that all of the Mg ii absorbing CGM inside the virial radius may not originate in fragmentation and condensation of the hot coronal gas phase inside the cooling radius of galaxy halos. It is of interest that as the W_r(2796) absorption threshold is increased, the difference between the covering fractions inside and outside the cooling radius decrease, such that for W_cut = 1.0 Å, they are virtually identical with f_c(η_c) ≃ 0.2. This indicates that the more optically thick and or kinematically complex the material is, the more uniformly distributed it is with respect to the theoretical cooling radius. We further note that this trend is not sensitive to virial mass.

In Section 3.1 we investigated, R(M_h), the absorption radius, and the degree to which it shows dependence on virial mass for various W_r(2796) absorption thresholds (see Figure 1). The results indicate that when the W_r(2796) absorption threshold is small, W_r(2796) ⩾ 0.1 Å, the absorption radius is highly proportional to virial mass (γ ≃ 0.45) and the covering fraction is quite large f_c ≃ 0.8. As the absorption radius is examined for progressively stronger absorption, the proportionality to virial mass progressively decreases, such that for W_r(2796) ⩾ 1.0 Å, γ ≃ 0.2. In addition, the covering fraction decreases to f_c ≃ 0.35.

For W_r(2796) ⩾ 0.1 Å, the fitted absorption radius increases by an order of magnitude (from ∼10 kpc to ∼200 kpc) over four decades of virial mass (10 ⩽ log M_h/M_☉ ⩽ 14). As such the fit predicts very extended absorption (D > 200 kpc) for log M_h/M_☉ > 13 when the weakest absorption is included. Since we do not probe impact parameters greater than D = 200 kpc, we do not have the data to verify this. Interestingly, in a statistical study of 50,000 Mg ii absorbers (0.4 ⩽ z ⩽ 2.5) compared with images of the quasar fields using SDSS data, Zhu et al. (2013) show that the mean W_r(2796) follows a decreasing power law with impact parameter out to 10 Mpc. At D = 100 kpc, the mean equivalent width of their sample is W_r(2796) ≃ 0.2 Å (comparable to our mean equivalent width at this impact parameter), and is W_r(2796) ≃ 0.003 Å at D = 10 Mpc. Assuming Navarro–Frenk–White (NFW) density profiles, they find that the surface density profile of Mg ii absorbing gas, Σ(Mgii) (M_☉ pc⁻²) is dominated by the single halo term out to 1 Mpc, outside of which the two-halo term dominates the gas surface density profile. The results of Zhu et al. (2013) corroborate the idea that Mg ii absorption can be highly extended for weaker absorption and may have implications for understanding the redshift path density of the population of weak Mg ii absorbers (Churchill et al. 1999; Rigby et al. 2002; Narayanan et al. 2007; Evans et al. 2013).

For W_r(2796) ⩾ 1.0 Å, our fitted relation predicts that for the most optically thick and/or kinematically complex absorbing gas, the sensitivity of the absorption radius to virial mass is not as pronounced, such that the radius increases by no more than a factor of five (from ∼30 kpc to ∼150 kpc) over the range 10 ⩽ log M_h/M_☉ ⩽ 14.

The decrease in both the slope, γ, and the normalization, R_*, with increasing W_r(2796) absorption threshold is strongly governed by the fact that the average covering fraction decreases for stronger absorption and for increasing D (see Nielsen et al. 2013b, and references therein). Most notably, it may be the differential behavior with both M_h and W_r(2796) absorption threshold in the "rate" at which the covering fraction, f_c(D), decreases with impact parameter (i.e., the slope of f_c(D)) that governs the behavior of γ and R_* with W_r(2796) absorption threshold (see Figure 2 of Churchill et al. 2013 for an illustration of this differential behavior in f_c(D)).

For the lowest (highest) W_r(2796) absorption threshold, the relatively steeper (shallower) virial mass dependence of R(M_h) reflects the steeper (shallower) decline in f_c(D). Note that this possible effect is most pronounced in the regime log M_h/M_☉ < 12, because the change in the slope of f_c(D) with W_r(2796) absorption threshold is most pronounced for lower virial mass galaxies, being steepest for the lowest W_r(2796) absorption threshold. This latter fact, in particular, results in the relatively large value of γ for the absorption threshold W_cut = 0.1 Å. Note that the differential behavior in f_c(D) is naturally explained by the self-similarity of the Mg ii absorbing CGM with virial mass, as discussed in Churchill et al. (2013).

The substantial uncertainties in R_* reflect the degree of fuzziness (both radially and spatially) in the mean absorption radius for a given absorption threshold. As such, the parameterizations of the absorption radius with virial mass reflect a correlation between impact parameter and virial mass with the interpretation that, on average, higher mass halos have a more extended CGM with lower geometric covering fractions.

We caution that the formalism of an absorption radius does not imply a well-defined boundary to the extent of the absorbing gas. The parameterization itself, as applied in this work, incorporates the assumptions of a spherical radius (circular in projection) and that the sky covering of the absorbing material is random. In hydrodynamic cosmological simulations, asymmetric filamentary structure in the cool/warm phase of the CGM is a common feature of simulations (Kereš et al. 2005, 2009; Dekel & Birnboim 2006; Ocvirk et al. 2008; Dekel et al. 2009; Ceverino et al. 2010; van de Voort et al. 2011; van de Voort & Schaye 2012; Goerdt & Burkert 2013). Kacprzak et al. (2010) showed that much of the Mg ii absorbing gas in the outer regions of the CGM is in the form of filaments. Using 123 galaxies from the MAGiiCAT sample, Kacprzak et al. (2012) reported that the covering fraction is a maximum f_c = 0.80 along the projected minor axis, is f_c = 0.65 along the projected major axis, and minimizes at f_c = 0.50 at projections intermediate to these two galactic axes. Thus, for Mg ii absorbing gas, the assumptions of a spherical geometry and random covering fraction are not supported by simulations nor observations so that the above parameterization provides the mean behavior of the Mg ii-absorbing CGM with virial mass averaged over all galaxy orientations.

By scaling the absorption radius by virial radius, we obtain the remarkable result that the mass-normalized absorption envelope, η(M_h), is very weakly dependent on virial mass (γ' ⩽ ±0.1) and has a value of $\eta _{\rm v}^{\ast } = R_{\ast }/R^{\ast }_{\rm vir} \simeq 0.3$ independent of W_r(2796) absorption threshold. Inspection of Figure 2 reveals that the number of galaxies with absorption above all W_r(2796) absorption thresholds drops dramatically outside the inner 30% of the virial radius. This suggests that both optically thin and/or kinematically quiescent and optically thick and/or kinematically complex absorbing gas is strongly concentrated within the inner 30% of the virial radius regardless of the virial mass of the galaxy.

The relatively weak dependence of η(M_h) on M_h and the decreasing covering fraction with increasing W_r(2796) absorption threshold are both consistent with the fact that the slope of f_c(D/R_vir) is virtually identical for lower and higher virial mass galaxies, but becomes shallower with increasing W_r(2796) absorption threshold (see Figure 2 of Churchill et al. 2013 for an illustration of this behavior in f_c(D/R_vir)).

The behavior of both the absorption radius and the mass-normalized absorption envelope are consistent with the results presented in Figures 4 and 5, which show that the mean W_r(2796) increases with virial mass within a finite impact parameter range, but is constant with η_v = D/R_vir, especially for η_v ⩽ 0.3. There is a remarkable self-similarity in the mean absorption relative to the virial radius over the full range of virial masses represented in our sample. The Mg ii column densities (governed by metallicity, density, and cloud size), and/or kinematics (number of clouds) are, on average, highly similar across virial mass within the inner 30% of the virial radius, and possibly out to the virial radius. However, since the virial radius is proportional to virial mass, the physical extent of the absorbing gas is greater for galaxies with higher virial mass. For the CGM to have the similar average W_r(2796) for all virial masses as a function of η_v = D/R_vir, it implies that the mean W_r(2796) increases with virial mass in finite impact parameter ranges (which is confirmed with our sample).

The observed increase in the mean W_r(2796) with virial mass in finite range of impact parameter is also apparent in the virial mass dependence of the upper envelope of absorption, as shown in Figure 3(a). This implies a virial mass "gradient" in the W_r(2796)–D plane in the direction of increasing W_r(2796). The steepening in the relationship between W_r(2796) and virial mass as impact parameter is decreased, as shown in Figure 3(b), reflects the fact that this mass gradient is steeper at smaller impact parameters.⁶ To a large degree, this mass gradient provides insight into the systematic scatter of W_r(2796) on the W_r(2796)–D plane, and the significant reduction of scatter of W_r(2796) on the W_r(2796)–η_v plane shown in Churchill et al. (2013). Since, on average, virial mass is correlated with galaxy luminosity (per the formalism of halo abundance matching), this explains the significant systematic luminosity segregation on the W_r(2796)–D plane reported in Paper II (Nielsen et al. 2013b).

Thus, examination of the data using several methods, as presented in Figures 1–5, all corroborate a picture in which the Mg ii-absorbing CGM is self-similar with relative location with respect to the virial radius.

We have investigated the behavior of W_r(2796) with η_c = D/R_c, the projected location where the absorption arises with respect to the theoretical cooling radius. Within the theoretical formalism of the cooling radius, i.e., to the degree that it can be viewed as a truly physical phenomenon associated with galaxy halos, we find that Mg ii absorption is found both inside and outside the cooling radius. For the optically thin and/or kinematically quiescent gas, the covering fraction is roughly a factor of two higher inside the cooling radius as compared to outside. However, for stronger absorption, the covering fraction is independent of whether the gas is outside or inside the cooling radius.

The model of Maller & Bullock (2004) does not predict cool/warm clouds in the CGM outside the cooling radius. If the Mg ii absorbing clouds have a single origin of fragmentation and condensation out of the hot coronal gas, then our findings might suggest the underlying physical principles from which a theoretical cooling radius is derived should be questioned. However, the model of Maller & Bullock (2004), by design, does not include stellar feedback mechanisms nor accretion from the IGM or mergers.

If there is reality to the theoretical cooling radius, we would then infer that absorbing structures residing outside the cooling radius are either recycled/processed clouds, winds, and/or infalling material and that the properties of cool/warm CGM gas are not fundamentally governed by where the gas resides relative to the cooling radius. That is, the non-negligible covering fraction for Mg ii absorption outside the theoretical cooling radius corroborates a multiple origins scenario for the cool/warm CGM provided by direct observation of winds (Tremonti et al. 2007; Martin & Bouché 2009; Weiner et al. 2009; Rubin et al. 2010; Martin et al. 2012), infall (Rubin et al. 2011; Kacprzak et al. 2010, 2012), rotation kinematics (Steidel et al. 2002; Kacprzak et al. 2011), superbubble kinematics (Churchill et al. 1995; Bond et al. 2001; Ellison et al. 2003), and orientation effects (Bordoloi et al. 2011; Bouché et al. 2011; Kacprzak et al. 2012).

Bouché et al. (2006) used a large (≃ 1800) sample of z ≃ 0.5 Mg ii absorbers with W_r(2796) > 0.3 Å and some 250,000 luminous red galaxies (LRGs) to obtain a statistical relation between equivalent width and virial mass for their flux-limited LRG sample. They estimated the virial masses of the absorbers by measuring the bias in the absorber–LRG cross-correlation relative to the LRG auto-correlation function. They reported a 3σ anti-correlation between virial mass and equivalent width, which they interpret as showing that Mg ii absorbers are not virialized within their host halos but instead originate from galactic winds in star forming galaxies.

In a study using a similar sample size and covering a similar redshift range, Gauthier et al. (2009) applied an essentially identical method and report a ∼1σ anti-correlation between equivalent width and virial mass for their volume-limited LRG sample. Defining stronger absorbers to have W_r(2796) > 1.5 Å and weaker absorbers to have 1.0 < W_r(2796) ⩽ 1.5 Å, they conclude that their weaker Mg ii absorbers (associated with log M_h/M_☉ < 13.4) are clustered more than their stronger absorbers (associated with log M_h/M_☉ < 12.7). Such a result would be consistent with the halo occupation model of Tinker & Chen (2008) in which the strongest Mg ii absorbers are suppressed in the most massive halos, which would reduce the clustering of strong Mg ii absorbers.

Motivated by the model of Tinker & Chen (2008) and the W_r(2796)–M_h anti-correlation of Bouché et al. (2006), Lundgren et al. (2009) also undertook a similar analysis of 0.36 ⩽ z ⩽ 0.8 Mg ii absorbers with W_r(2796) > 0.8 Å, which they cross correlated with some 1.5 million LRGs (volume-limited sample). They report a "marginal" anti-correlation (significance level not stated) between W_r(2796) and virial mass and conclude that their weaker Mg ii absorbers occupy halos some 25 times more massive than their stronger absorbers. With a substantially larger and more controlled sample than that of Bouché et al. (2006), they were unsuccessful at obtaining a higher significance in the W_r(2796)–M_h anti-correlation; they actually found a weaker signal than Bouché et al. (2006). The Lundgren et al. (2009) work actually calls into question the veracity of a W_r(2796)–M_h anti-correlation.

In direct conflict with the 3σ result of Bouché et al. (2006) and the less statistically significant follow-up results of Gauthier et al. (2009) and Lundgren et al. (2009), we find that our data are highly consistent with no correlation between W_r(2796) and M_h when all impact parameters are considered. As shown in Appendix C, a rank correlation test is highly consistent with the null hypothesis of no correlation (0.1σ).

In Figure 8(a), we directly compare our sample to those of Bouché et al. (2006), Gauthier et al. (2009), and Lundgren et al. (2009). Since Bouché et al. (2006) reports the most significant W_r(2796)–M_h anti-correlation, we binned our data to match theirs, however, we plot the Gauthier et al. (2009) and Lundgren et al. (2009) data as presented in those works. For absorbers with W_r(2796) > 0.3 Å, our results are consistent within uncertainties with all three absorber–LRG cross-correlation studies. Interestingly, there is a slight discrepancy at W_r(2796) ∼ 2.4 Å where Bouché et al. (2006) obtain an upper limit on the mean virial mass that is ∼2σ lower than our value. Since the Bouché et al. (2006) sample is limited to D ⩽ 140 kpc, we recomputed <log M_h/M_☉ > for the subsample of our data for which D ⩽ 140 kpc. These points are plotted as open points. There is no change in the mean virial mass for W_r(2796) > 2 Å and only an insignificant reduction in the mean virial mass for W_r(2796) < 2 Å.

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Since we probe well below W_r(2796) = 0.3 Å, where the majority of our equivalent width measurements are upper limits (3σ), we computed <log M_h/M_☉ > for the non-absorbers and the weakest absorbers. In Figure 8(b), we compare our data directly to those of Bouché et al. (2006) and add the data point for W_r(2796) < 0.3 Å absorbers and non-absorbers. Our data are consistent with no W_r(2796)–M_h anti-correlation over this much broader equivalent width range. We caution that ≃ 20% of the W_r(2796) < 0.3 Å absorbers and non-absorbers reside at D > 140 kpc, whereas the Bouché et al. (2006) systems all reside at D < 140 kpc. However, as we showed above, the inclusion of larger impact parameters has a negligible effect on the mean virial mass.

In summary, we find no anti-correlation between W_r(2796) and M_h, whether we binned our data to obtain the means (per Figure 8) or performed statistical tests on the unbinned data. In fact, we find trends that suggest W_r(2796) and M_h are correlated in finite ranges of impact parameter. Our data, as presented in Figure 8, are statistically consistent with the cross correlation clustering analyses of Bouché et al. (2006), Gauthier et al. (2009), and Lundgren et al. (2009); however, our results do not imply any suggestion of a W_r(2796)–M_h anti-correlation.

If, per the halo occupation model of Tinker & Chen (2008), higher mass halos have a CGM environment that suppresses Mg ii absorbers, then the covering fraction of the CGM would be observed to decline for higher virial mass galaxies. This would reduce the clustering of stronger absorbers. However, the model of Tinker & Chen (2008) is tuned to match the W_r(2796)–M_h anti-correlation reported by Bouché et al. (2006).

In this work, we have shown that the data do not support a W_r(2796)–M_h anti-correlation,⁷ but do support a strong positive trend in finite impact parameter ranges. Churchill et al. (2013) showed that the covering fraction is effectively invariant with virial mass (see their Figures 3 and 4), even for different W_r(2796) absorption thresholds. This fact places tension on the halo occupation model, and it remains to be worked out how this would change the argument for weaker clustering of stronger Mg ii absorbers.

We expect that the lack of a W_r(2796)–M_h anti-correlation and the invariance in the Mg ii covering fraction with virial mass would nullify a weaker clustering of stronger Mg ii absorbers. Interestingly, Rogerson & Hall (2012), using paired quasar sightlines of Mg ii absorbers, were able to show that the Tinker & Chen (2008) model was ruled out because the model failed to reproduce the observed variation in Mg ii absorption between sightlines.

In addition to quantifying the extent of Mg ii absorbing gas, previous works have also investigated physical interpretations to understand the CGM, both within the context of other halo gas statistics and the theoretical framework of gas accretion and galactic outflows. Along these lines, Bouché et al. (2006) claim that an anti-correlation between W_r(2796) and M_h is a natural consequence of the behavior of absorber statistics and that since there are no strong Mg ii absorbers at large distances from galaxies, equivalent width must be inversely proportional to virial mass. They argue that this physical relation follows from the combined results that the absorption radius is proportional to galaxy luminosity, R(L) = R_*(L/L*)^β with β > 0, and that the equivalent width is inversely proportional to absorption radius, W_r(2796) = α₁log R(L) + α₂ (with α₁ < 0), the latter reflecting the W_r(2796)–D anti-correlation.

The Bouché et al. (2006) argument is based upon the assumption that there is no virial mass dependence on the slope and normalization (α₁ and α₂) of the upper absorption envelope (absorption radius), which forces the inference that there is a horizontal virial mass gradient on the W_r(2796)–D plane such that higher mass halos are preferentially found at larger impact parameter. As shown in Figure 3(a), a galaxy with strong Mg ii absorption usually has absorption at small impact parameter. However, the fact that absorption occurs closer to a galaxy does not imply that the halo is low mass and the fact that absorption occurs far from a galaxy does not imply that it has high virial mass. What we find is that stronger (weaker) absorption in a finite impact parameter range implies the host galaxy has a higher (lower) virial mass, and this applies regardless of whether the impact parameter is small or large (also see Figure 3(b)).

Galaxies that inhabit more massive dark matter halos simply have stronger absorption at a given distance, even though W_r(2796) decreases with distance at a given mass. Our data directly show that the virial mass gradient on the W_r(2796)–D plane is vertical, such that the slope of the upper absorption envelope steepens with increasing virial mass. This behavior results in a flat relationship between the mean W_r(2796) as a function of virial mass when all impact parameters are included in the mean.

Since the halo abundance matching method yields the average virial mass for a galaxy with a measured M_r, the behavior of the Mg ii absorbing CGM reported in this work reflects an averaged behavior with virial mass, and is not based upon a 1:1 correspondence between W_r(2796) and a dynamically measured virial mass, M_h. However, we note several reasons that the averaged behavior is an accurate representation.

First, the analysis does provide a 1:1 correspondence between W_r(2796) and M_r. Each galaxy associated with a measurement of Mg ii absorption has a measured r-band luminosity. Second, for halo abundance matching, the relationship between the average virial mass and M_r at a given redshift is monotonic and smooth (see Figure 9); thus, the main difference between results obtained using the average virial mass as opposed to the measured M_r is contained in the slope of the M_h–M_r relation as a function of M_r. This slope steepens at the bright end, which effectively provides a stretching in the dynamic range of the galaxy property being compared to the Mg ii absorption, essentially increasing the leverage and/or moment arm over which the CGM–galaxy connection can be explored. Third, the general behavior of W_r(2796) with virial mass, including mass dependence of the covering fraction, f_c(D), mass segregation on the W_r(2796)–D plane, etc., is also seen directly with $L_B/L_B^{\ast }$, $L_K/L_K^{\ast }$, M_B, and M_K (Nielsen et al. 2013b). For the luminosities, the trends with M_K (a proxy for stellar mass) are invariably the most statistically significant.

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The higher statistical significance for these trends and correlations with virial mass is a consequence of the rapid increase in virial mass at the bright end of the LF, which, as stated above, provides added leverage for exploring the galaxy–CGM connection. The added benefit of employing the average virial mass is that the average virial radius of an M_r galaxy can be incorporated into the analysis. Knowing the physical extent of the dark matter halo has provided enhanced insight. For example, the significantly reduced scatter and vanishing of virial mass segregation on the W_r(2796)-η_v plane as compared to the W_r(2796)-D plane, the tight scaling of W_r(2796) with (η_v)⁻², the self-similarity of the covering fraction, f_c(η_v), and the invariance of the mean W_r(2796) with virial mass in finite η_v ranges. Virial mass also provides a formalism for estimating the theoretical cooling radius, which has provided some insight into the fact that Mg ii absorption strength and covering fraction shows no dependence on this theoretically based quantity.

A main result of this work is that the properties of the cool/warm component of the CGM are self-similar with virial mass and fundamentally connected to the parameter η_v = D/R_vir, the projected galactocentric distance of the gas relative to the virial radius. Regardless of viral mass, the mean Mg ii absorption strength is first and foremost governed by where it resides with respect to the virial radius of the halo. We found that the majority of the Mg ii absorbing cool/warm CGM is located within the inner 30% of the virial radius.

Though the mean W_r(2796) strongly trends toward a positive correlation with virial mass in finite impact parameter ranges, the overall lack of a correlation between the mean W_r(2796) and virial mass when all impact parameters are included is due to the highly significant anti-correlation between W_r(2796) and impact parameter at fixed virial mass. Most remarkable is that the mean W_r(2796) is constant as a function of η_v = D/R_vir for η_v ⩽ 0.3, and may be constant all the way out to the virial radius.

Nielsen et al. (2013b) showed that the Mg ii covering fraction decreases with increasing W_r(2796) absorption threshold at all impact parameters and decreases with increasing distance from the central galaxy. Churchill et al. (2013) showed that the Mg ii absorption covering fraction is effectively invariant as a function of virial mass for all W_r(2796) absorption thresholds, though it decreases as the W_r(2796) absorption threshold is increased. The latter result places tension on the notion that "cold-mode" accretion is suppressed in higher mass halos (log M_h/M_☉ ⩾ 12) as purported by Birnboim & Dekel (2003), Kereš et al. (2005), Dekel & Birnboim (2006), and Stewart et al. (2011). One solution is that much of the Mg ii absorbing gas in higher mass halos arises in outflowing winds and/or infalling metal enriched sub-halos (low-mass satellite galaxies, some of which may be embedded in filaments).

Whatever the reasons that explain invariance of the covering fraction with galaxy virial mass, the data indicate that the mean W_r(2796) scales as an inverse-square power law, (D/R_vir)⁻², with remarkably low scatter over several decades of virial mass (Churchill et al. 2013). Combined, the virial mass invariance of the covering fraction and the inverse-square profile of the mean equivalent width place strong constraints on the nature of the low-ionization CGM and are highly suggestive that the CGM is self-similar with the virial mass of the host galaxy.

We note that our results are remarkably consistent with the findings of Stocke et al. (2013), who examined the CGM in multiple low- and high-ionization transitions for ≃ 70 galaxies at z ⩽ 0.2. They find that the majority of the metal-line absorbing gas in the CGM resides within the inner 50% of the virial radius. They also report that, once virial radius scaling is applied, there is little distinction between CGM clouds as a function of galaxy luminosity, radial location, or relative velocity. Furthermore, Stocke et al. (2013) find that the absorbing cloud diameters decrease with (D/R_vir)^{−1.7 ± 0.2}, which is very close to an inverse square relationship (however, the derived diameters exhibit considerable scatter about the relation).

Stating that the cool/warm CGM is self-similar across a wide range of virial mass is not equivalent to stating that the CGM is identical for all galaxies vis-à-vis a simple scaling with virial radius. Evidence for multiple origins of Mg ii absorbing clouds, such as winds (Tremonti et al. 2007; Martin & Bouché 2009; Weiner et al. 2009; Rubin et al. 2010; Martin et al. 2012; Bradshaw et al. 2013; Bordoloi et al. 2013), superbubbles (Churchill et al. 1995; Bond et al. 2001; Ellison et al. 2003), infall (Rubin et al. 2011; Kacprzak et al. 2010, 2012), and evidence for rotation kinematics (Steidel et al. 2002; Kacprzak et al. 2011) and orientation dependencies (Bordoloi et al. 2011; Bouché et al. 2011; Kacprzak et al. 2012) precludes such a notion. Furthermore, galaxies of different dark matter halo masses live in different overdensities and therefore local environments.

We are faced with the central question: how is it that the various physical processes governing the cool/warm baryons in the CGM, which respond to their host dark matter density profile and local over-dense environment, yield mass invariant covering fractions and self-similar radial profiles of the mean Mg ii absorption strengths with location relative to the virial radius, R/R_vir, over a large range of galaxy virial mass?

It is well established that higher mass halos live in higher overdensity regions and are surrounded by greater numbers of sub-halos (e.g., Mo & White 1996; Klypin et al. 2011). The more massive sub-halos can form stars and chemically enrich their immediate surrounding, such that absorbing gas in sub-halos likely comprises some component of what we call the cool/warm CGM of higher mass halos (Kepner et al. 1999; Gnat & Sternberg 2004; van de Voort & Schaye 2012). Lower mass halos, in contrast, live in lower overdensity regimes, where the contribution of enriched gas to their CGM from sub-halos is presumably lower. On the other hand, the influence of stellar feedback may have a relatively more important influence on the CGM of lower mass galaxies (e.g., Dalla Vecchia & Schaye 2008; Weinmann et al. 2012; Ceverino et al. 2013; Trujillo-Gomez et al. 2013). A great deal of theoretical work is pushing the frontiers of our understanding of the different dominating physical processes governing the ISM–CGM–IGM cycle as a function of galaxy stellar and virial mass, and we will highlight some of these in the below discussion.

As speculated by Werk et al. (2013), the decrease in the average absorption strength in low-ionization metals with increasing impact parameter could imply a decreasing surface density in these ions, a decreasing metallicity, and/or an increasing ionization state with increasing galactocentric distance, R, from the central galaxy. Interestingly, Stocke et al. (2013), having performed ionization modeling of CGM absorbing clouds for their sample, find no clear trends in cloud ionization parameter, density, metallicity, or temperature with distance from the central galaxy or with galaxy luminosity (though they do find trends for decreasing cloud sizes and masses with distance from the central galaxy, which reflects the decreasing H i column density). This would suggest that neither a metallicity gradient nor an ionization gradient in the CGM is the driving mechanism governing the W_r(2796)–D anti-correlation out to the virial radius.

Werk et al. (2013) also conclude that it is unlikely that the star formation rate in the central galaxy is a dominant factor governing the strength of low ion absorption. Instead, they find that the column densities of the low-ionization species correlate with stellar mass, indicating that there is more circumgalactic gas in more massive galaxy halos. Similarly, Zhu & Ménard (2013) find that the amount of Ca ii in halos is larger for galaxies with higher stellar mass. Since stellar mass and virial mass are correlated, these results are consistent with our findings of a more extended upper envelope to the absorption with increasing virial mass (see Figure 3(a)) and a proportionality between W_r(2796) and virial mass in finite impact parameter ranges (see Figure 4).

Examining the average properties z = 0.25 galaxies in low-resolution cosmological simulations that include momentum-driven winds, Ford et al. (2013b) employed ionization modeling of the CGM gas and found that Mg ii absorbing gas column density is centrally concentrated and decreases with increasing R from the central galaxy with very little qualitative morphological difference in the radial profiles with halo mass (filaments, satellite galaxies, and sub-halos are azimuthally smoothed out). They find virtually no temperature gradient with impact parameter for the low-ionization species, which arises in T = 10⁴–10^4.5 K gas. Though more massive halos have larger hot gas fractions (Kereš et al. 2005; van de Voort & Schaye 2012), Ford et al. (2013b) find the overdensity of gas where H i absorption arises increases with halo mass from Δρ/ρ = 10² for log M_h/M_☉ = 11 to Δρ/ρ = 10³ for log M_h/M_☉ = 13 (D ∼ 100 kpc); most of the H i arises in cool/warm T < 10⁵ K gas, even in the highest mass halos.

The simulations and modeling of Ford et al. (2013b) clearly show that the extent of the cool/warm CGM increases with increasing galaxy virial mass; galaxies in more massive halos have a more extended CGM than galaxies in lower mass halos. Inspection of their Mg ii column density profiles show a predicted mean Mg ii column density of <N(Mgii)> = 10¹³ cm⁻² at D ≃ 20 kpc for log M_h/M_☉ = 11 and D ≃ 100 kpc for log M_h/M_☉ = 13. Assuming thermal broadening for T = 10^4.5 K, this corresponds to a mean absorption strength of <W_r(2796) > =0.15 Å, a value consistent with our measurements at these impact parameters. Thus, the simulations and ionization modeling predict that a given W_r(2796) value, on average, will be measured at a larger impact parameter in higher mass halos in a manner that is consistent with our findings. We also note that these example points probe the same η_v = D/R_vir, since the ratio of the virial radii of the two simulated galaxies, (10¹¹/10¹³)^1/3 ≃ 1/5, equals the ratio of the impact parameters of the respective galaxies. A constant mean W_r(2796) with η_v is precisely the behavior we showed in Figure 5.

As such, the results of Ford et al. (2013b) provide a natural explanation for the self-similarity of the cool/warm CGM with virial mass. Their simulations imply that winds contribute to the presence of Mg ii absorption in the CGM such that higher mass halos have larger mean W_r(2796) at a fixed impact parameter. Sub-halos (satellites) and filamentary infall play a role as well, since these contributions were simply smoothed (averaged out) in the radial gas profiles presented by Ford et al. (2013b). That is, predominantly photoionized outflowing winds and accreting sub-halo clumps could go a long way toward explaining the average self-similar properties of the low ionization CGM across several decades of virial mass.

It would be of interest to examine the column density profiles from the Ford et al. (2013b) work as a function of D/R_vir and virial mass to quantify the degree that they are self-similar with virial mass. The foremost observational constraints that any model would be required to satisfy are the inverse-square profile, W_r(2796)∝(D/R_vir)⁻², and the virial mass invariant covering fraction that decreases with increasing W_r(2796) absorption threshold, both of which should be measured using "mock" absorption line analysis through the simulated halos.

Though our data show that the average Mg ii absorption strength in the CGM obeys clear trends with virial mass, impact parameter, and virial radius, there is a great deal of spread in the distribution of W_r(2796). The spread is significant enough that the frequency of non-detections increases with increasing impact parameter. This could possibly be due to substantial variations in the metallicity of the CGM from sightline to sightline, even in the absence of a clear metallicity gradient out to the virial radius. We now consider the possibility of a correlation between stellar/virial mass with the metallicity of the cool/warm CGM in gas and plausible explanations for strong variations in metallicity in the CGM from sightline to sightline.

Tremonti et al. (2004) reported a tight correlation between stellar mass, M_*, and gas-phase metallicity, Z_ISM, of the ISM spanning three decades of stellar mass and a factor of 10 in metallicity. Mannucci et al. (2010) examined the more general relation between stellar mass, gas-phase metallicity, and star formation rate (SFR) and found a tight surface in this three-dimensional space, dubbed the fundamental metallicity relation (FMR). At low stellar mass, metallicity decreases with increasing SFR, while at high stellar mass, metallicity is independent of SFR. Bothwell et al. (2013) showed that SFR may not be the fundamental third parameter of the FMR; they find that H i mass drives the stellar-mass metallicity relation such that metallicity continues to correlate with stellar mass as H i mass increases. Stellar mass and H i mass correlations with gas phase metallicity in the ISM may suggest a similar relation, on average, in the CGM. However, the CGM being much more extended, and being an interface with the IGM, is undoubtedly more complex such that sightline to sightline variations could mask a galaxy/halo mass metallicity correlation.

Indeed, in order to understand the stellar-mass metallicity correlations, the flow and recycling of ISM and IGM gas through the CGM must be invoked and tuned. Davé et al. (2011) determined that gas content is regulated by a competition between inflow and gas consumption within the ISM, which is governed by the star formation law. That is, star-forming galaxies develop via a slowly evolving equilibrium balanced by inflows (driven by gravity/mass), wind recycling, star formation rates, and outflows, the latter regulating the fraction of inflow that gets converted into stars (Davé et al. 2011). Dayal et al. (2013) found that for more massive galaxies, ISM metal enrichment due to star formation is diluted by inflow of metal-poor IGM gas that yields a constant value of the ISM gas metallicity with SFR (thereby reproducing the FMR at high mass). In these massive galaxies, the effects of outflows are severely mitigated due to the deep gravity wells. Conversely, lower mass galaxies, which have smaller SFR, produce lower metallicity outflows, but they are more efficiently distributed throughout the CGM due to the shallower potential wells. A similar model by Lilly et al. (2013) indicates that the M_*–M_h relation, established by baryonic processes within galaxies, suggests a significant fraction (40%) of baryons coming into the halos are being processed through the galaxies.

Thus, we see that the mass–metallicity relationships of galaxies (e.g., Tremonti et al. 2004; Mannucci et al. 2010; Bothwell et al. 2013) theoretically suggest a regulatory physical cycle between the ISM and the CGM that involves lower metal enrichment of the CGM in lower mass galaxies and higher metal enrichment of the CGM in higher mass galaxies; however, the wind/outflowing material is more efficiently distributed into the CGM in lower mass galaxies and less efficiently distributed in higher mass galaxies. This general behavior of the wind/recycled gas, coupled with the rates at which clumpy and filamentary accretion is mixed in the CGM, likely provides an excellent first-order physical understanding of how the CGM of galaxies living in different mass halos can be self-similar in their mean Mg ii absorption properties. We remind the reader that self-similar means the projected profile of Mg ii absorption strength with respect to the virial radius is universal, i.e., W_r(2796)∝(D/R_vir)⁻², and the covering fraction of the CGM is independent of virial mass.

Turnshek et al. (2005) reported that gas-phase metallicity strongly correlates with the velocity spread of Mg ii (W_r(2796) expressed in velocity units) for large N(Hi) absorbers. Using zCOSMOS galaxies in the redshift interval 1.0 ⩽ z ⩽ 1.5, Bordoloi et al. (2013) report that the Mg ii equivalent width of the outflowing component increases with both galaxy stellar mass and star formation rate. At similar stellar masses, the blue galaxies exhibit a significantly higher outflow equivalent width as compared to red galaxies. In the UKIDSS Ultra-Deep Survey, Bradshaw et al. (2013) found that the highest velocity outflows are found in galaxies with the highest stellar masses and the youngest stellar populations. They conclude that high-velocity galactic outflows are mostly driven by star-forming processes consistent with a mass–metallicity relation.

On the other hand, Lehner et al. (2013) reported a bimodality in the metallicity of the CGM of luminous galaxies and conclude that the more metal-rich absorbers likely originate from the nearby large galaxy in the form of outflowing or recycling gas while the lower metallicity gas is infall from the IGM. Interestingly, of the galaxies for which Stocke et al. (2013) could constrain the cool/warm CGM metallicities, nine absorbers have Z_CGM ≃ Z_ISM and velocity offsets, Δv, from the galaxies that are ∼10% of the halo escape velocity, v_esc; these are identified as bound clouds, possibly recycling material. Five absorbers have Z_CGM ≃ Z_ISM and velocities indicating Δv > v_esc; these are identified as unbound outflows. Three absorbers have Z_CGM ⩽ 0.2 Z_ISM and are identified with infall. In several cases geometrical constraints confirm the flow direction of the studied clouds. Stocke et al. (2013) find no discernible differences in the densities, ionization parameters, cloud sizes or masses between the inflowing and outflowing absorbers.

In summary, the stellar mass gas-phase metallicity correlations places strong constraints on the outflow and recycling of metal-enriched gas, the inflow of metal-poor gas, and the incomplete mixing of the gas through the CGM, while absorption line observations show there are variations in gas metallicity that are consistent with these physical processes. The theory and observations indicate a connection between the mean W_r(2796), galaxy stellar mass, and gas phase metallicity and kinematics. Higher metallicity gas in chemically processed wind material gives rise to larger equivalent widths in absorption. Conversely, unmixed inflow material gives rise to smaller absorption equivalent widths due to the lower metallicity.

One would then expect the scatter in the W_r(2796) distribution to be primarily due to metallicity variations from sightline to sightline (recalling little variation in cloud ionization, temperature, and density from ionization models). The decrease in Mg ii covering fraction with increasing distance from the central galaxy (and absorption threshold) would then quantify the scatter in the metallicity of the cool/warm CGM and imply that the metallicity is more uniformly distributed from cloud to cloud at smaller galactocentric distances but highly variable from absorbing cloud to absorbing cloud at larger galactocentric distances. If this scenario is correct, it could partially explain why the frequency of sightlines with upper limits on W_r(2796) is higher at larger impact parameters.

Further insight is gleaned from the simulations of van de Voort & Schaye (2012), who conducted a thorough study of the CGM parameter space with virial mass, stellar feedback, and distance from the central galaxy.⁸ For galaxies in log M_h/M_☉ ≃ 12 halos, their "cold" CGM gas exhibits four orders of magnitude spread in metallicity at R ≃ R_vir, with density-weighted mean Z_gas ≃ 0.01, while the "hot" gas has only a single order of magnitude spread with Z_gas ≃ 0.1. Deeper inside the virial radius at R ≃ 0.1R_vir, the metallicity of the inflowing gas has the narrow range 0.1 ⩽ Z_gas ⩽ 1; this spread broadens to lower metallicities, 10⁻² ⩽ Z_gas ⩽ 0.3, by R ≃ 0.3 R_vir and to 10⁻⁴ ⩽ Z_gas ⩽ 0.5 by R ≃ R_vir. On the other hand, the outflow metallicity spread remains at a constant 0.1 ⩽ Z_gas ⩽ 1 with radius out to R ≃ R_vir. The outflow fraction is ≃ 0.4, holding constant out to R_vir = 1.0 (mass weighted). Most of the gas mass is in the inflow. Across virial mass over the range 10 ⩽ log M_h/M_☉ ⩽ 13, just inside the virial radius, the outflow metallicity remains constant with galaxy virial mass within the range 0.03 ⩽ Z_gas ⩽ 0.5. However, the lower envelope on this large range in the inflow metallicity rises to a higher minimum metallicity as mass increases (the spread narrows toward a higher mean metallicity).

To the degree that Mg ii absorption probes the CGM, the general increasing spread in the metallicity with increasing R found by van de Voort & Schaye (2012) would be consistent with a growing frequency of sightlines with upper limits on W_r(2796) as the CGM is probed closer to the virial radius. This is consistent with a covering fraction that decreases with increasing distance from the central galaxy.

The behavior of the mass-normalized absorption envelope, η(M_h), remains to be understood. This envelope has a mean value of $\eta _{\rm v}^{\ast } \simeq 0.3$ (see Figure 2), is weakly dependent upon virial mass, and is independent of absorption threshold. This could be explained by a narrower range of metallicity within R ≃ 0.3 R_vir, as suggested by the simulations of van de Voort & Schaye (2012). Beyond this scaled radius, the spread in the metallicity increases, and the mean column density of Mg ii absorbing gas has declined (Ford et al. 2013b), which could be due to decreasing cloud sizes as impact parameter increases (Stocke et al. 2013). A second possibility is that wind material, whether bound or unbound, remains in an ionization state that is detectable in Mg ii absorption primarily within R = 0.3 R_vir. However, the lack of an ionization gradient in the cool/warm CGM gas studied by Stocke et al. (2013), and the simulations of Ford et al. (2013b), do not support this idea. A third possibility is that the majority of the Mg ii absorbing gas is bound and recycles such that the gas is confined within a "turnaround" radius of R/R_vir ≃ 0.3. This would imply that wind material, on average, would be required to reach R/R_vir ≃ 0.3 regardless of galaxy virial mass. Though this is not entirely consistent with the findings of Stocke et al. (2013), who find higher metallicity unbound CGM clouds in several instances, it is commensurate with the results of Ford et al. (2013a), who find that the majority of gas associated with Mg ii absorption is "recycled accretion," meaning that, regardless of the origin of the gas, it will accrete onto the galaxy within the time span of ∼1 Gyr.

The low frequency of sightlines with large W_r(2796) found outside R/R_vir = 0.3, may, on average, be due to enriched sub-halos (satellites) surrounding the more massive galaxies. For R/R_vir > 0.3, the growing frequency of very weak Mg ii absorption clouds and sightlines along which upper limits on W_r(2796) are measured, may be due to lower metallicity infalling material that has not fully mixed with the recycling material inside the putative "turnaround" radius. Accounting for satellites (around the more massive galaxies) that enrich their local medium as they infall and accounting for "pristine" infalling filaments, a wide range of W_r(2796) could arise from infalling gas. Though the observational evidence is quite compelling that the majority of strong Mg ii absorbers arise from metal-enriched wind driven material (described above), some large W_r(2796) values at large impact parameters could be due to enriched infalling satellites in the more massive galaxy halos.

For the mean W_r(2796) to be constant with virial mass inside this putative "turnaround" radius, we would need to invoke physics that conspires to yield a degeneracy between gas metallicity, velocity spread, and ionization conditions as a function of galaxy virial mass. That is, in the final analysis, we should view the self-similarity of the cool/warm CGM with virial mass as a reflection of a global quasi-equilibrium regulation in which cool/warm cloud creation, destruction and/or recycling timescales, hydrodynamical physics, and external reservoirs of CGM gas balance so as to yield the simple result that, on average, the equivalent width of Mg ii absorbing CGM gas is strongly connected to galactocentric distance with respect to the virial radius, especially within the inner 30%.

Though speculative in nature and only a qualitative picture, the scenario we outline illustrates the possibility that the self-similarity of the Mg ii absorbing CGM with virial mass could result from multiple processes that are consistent with what is currently known about galaxies, the ISM, the CGM, and the local IGM and environment. Furthermore, this view of the CGM is one that is fully consistent with simulations and models (cf. Davé et al. 2011; Dayal et al. 2013; Lilly et al. 2013) that go far to explain a holistic interconnectedness between star formation, stellar feedback, galaxy stellar and virial mass, and the gas cycles of the ISM, CGM, and IGM as constrained by observations.

The results we found for Mg ii absorption should equally apply for cool/warm gas absorption from other low-ionization potential metallic species such as Si ii, C ii, and Fe ii. In fact, as we mention previously, Zhu & Ménard (2013) have presented evidence that Ca ii CGM absorption is stronger in galaxies with higher stellar masses. It would be of interest to study the metallicity and abundance ratios as a function of galactocentric distance relative and with respect to the virial radius in order to discern whether the abundance gradient is flat with η_v = D/R_vir, declines smoothly, falls off precipitously at some point, or exhibits increasing scatter with increasing distance from the central galaxy. However, considering the findings of Stocke et al. (2013) for a sample of ≃ 70 galaxies, such an analysis will likely require an order of magnitude increase in the number of quasar sightlines through the CGM environment of galaxies.

As mentioned in Section 2.2, there is scatter in the $V_c^{\rm max}$–M_h relation in the Bolshoi halo catalogs due to the different formation times of halos of a given mass. We treat this scatter by calculating the mean virial mass, M_h, within a fixed luminosity bin, ΔM_r, and assign the standard deviation as the statistical uncertainty in the average virial mass. We compute one-sided standard deviations to obtain insight into the asymmetry of the virial mass distribution within the ΔM_r bin; as such, we are not presenting formally proper statistical uncertainty measurements but are quantifying the degree of scatter and skew in the underlying distribution of M_h employed in obtaining the mean value.

Since the LF has variable slope with M_r − 5log h, the mean M_h will have a systematic dependence on the width of ΔM_r. The expectation is that the broader the bin size, the more M_h will be skewed toward smaller values due to the increased abundance of fainter galaxies in the LF. Since the LF slopes are different at different redshifts, this systematic skew will be different at each redshift. In addition, increasing the width of ΔM_r results in the inclusion of more halos being averaged, which affects the adopted uncertainties in M_h.

In Figure 9(a), we present the COMBO-17 r-band LFs based upon the Schechter function parameter fits of Wolf et al. (2003). To examine how the width of ΔM_r, affects the systematics of and scatter in M_h, we varied ΔM_r over the range 0.1 ⩽ ΔM_r ⩽ 0.4 and performed the halo abundance matching over the range −24 ⩽ M_r − 5log h ⩽ −16. In Figures 9(b) and (c), we present log M_hh⁻¹/M_☉ versus M_r − 5log h for ΔM_r = 0.1 and ΔM_r = 0.4, respectively. The solid curves are the mean M_h and the shaded regions are the one-side standard deviations of the distribution of virial masses in the bin ΔM_r. The redshift dependence is shown by the individual curves.

Consider Figure 9(b). Note that the minimum and maximum M_h are different for each redshift bin. The maximum virial mass, which increases with decreasing redshift, is dictated by the distribution of virial masses in the halo catalog. The increase in the maximum virial mass with decreasing redshift reflects virial mass growth evolution. The minimum mass is dictated by the completeness of the velocity function, $n(V_c^{\rm max})$, at small M_h, as discussed in both Trujillo-Gomez et al. (2011) and Klypin et al. (2011). The truncation of $n(V_c^{\rm max})$ is at brighter luminosity at higher redshift because of the steeper LF at high redshift, i.e., the minimum virial mass in the catalog gets assigned to a brighter galaxy.

Comparing Figures 9(b) and (c), we find that the adopted bin size of ΔM_r has virtually no effect on the scatter of each mass estimate and no more than a 0.3 dex systematic lowering of M_h for ΔM_r = 0.4 as compared to ΔM_r = 0.1 in the regime of M_r − 5log h < −23. This systematic is due to the steepness of the LF at the very bright end. For our methods, we find that M_h is sensitive to the width of the luminosity bin to no more than ≃ 0.35 dex for the highest masses when we also account for statistical uncertainties. We adopted ΔM_r ⩽ 0.1 for this work in order to minimize the effect of the variable slopes of the LF.

Since the abundance of dark matter halos is known to high precision in the ΛCDM cosmology, a substantial source of potential systematic uncertainty in the derived M_h could arise from systematic errors in the evolution of the measured LF. To examine the range of possible systematics in our adopted virial mass calculations, we explored variations in M_h under the presumption that evolution in the observed LF from z = 1 to z = 0 is dominated by systematic measurement errors.

To emulate systematics in the LF, we abundance matched to the observed LF over the range −24 ⩽ M_r − 5log h ⩽ −16 in each of the five redshift bins using only the z = 0.1 Bolshoi halo catalog. We thus evolve the LF while holding the halo population constant. We then performed the identical exercise using only the z = 1.0 Bolshoi halo catalog, thereby holding the halo population constant but with abundance matching to a different virial mass distribution (separated by ∼7 Gyr of cosmic time). The exercise emulates plausible systematics in the evolution of the LF. We also varied the width of the luminosity bin used for obtaining the mean M_h, illustrating the ΔM_r = 0.1 and 0.4 cases.

In Figure 10, we plot the percent difference,

$$\begin{equation} \Delta \% = 100 \cdot \left[ \frac{M_{\rm \, h}(z\!=\!0.1) - M_{\rm \, h}(z\!=\!1.0)}{M_{\rm \, h}(z\!=\!0.1)} \right] , \end{equation} \tag{ A1 }$$

between the M_h obtained with the z = 0.1 and z = 1.0 Bolshoi halo catalogs. The results are shown for each redshift bin of the LF (colored as in Figure 9). Figure 10(a) illustrates the ΔM_r = 0.1 exercise, and Figure 10(b), illustrates the ΔM_r = 0.4 exercise.

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The exploration indicates that no more than a 4% systematic difference in M_h is likely to be present in our adopted values over the redshift range of our study. The effect monotonically increases toward the bright end of the LF and is somewhat independent of the shape of the LF in this luminosity regime. There is also some dependence on the faint end slope of the LF in the range M_r − 5log h > −20 at the level of 1% percent difference. We should have no more than a Δ% = 4% systematic error at the highest mass end based upon the reasonable assumptions we have incorporated to model systematic uncertainty in LF evolution.

Given the metallicity dependence of R_c, there can be uncertainty in the size of the cooling radius for fixed virial mass. However, as we discuss below, observations and theory corroborate that the average metallicity of the hot gas of galaxy halos can be well approximated as Z_gas ≃ 0.1 for a large range of virial mass and galaxy morphological type.

Observations of hot halos of spirals and ellipticals indicate they both obey the same L_X–L_K (0.5–2.0 keV X-ray and K-band luminosity) relations and the same L_X–T_X relations, from which a common origin of hot coronal gas in both early and late type galaxies is inferred (Crain et al. 2010). The similar correlations arise because L_X, L_K, and T_X are all proportional to virial mass.

Comparing observations to their gimic simulations (Crain et al. 2009), drawn from the Millennium Simulation (Springel et al. 2005), Crain et al. (2013) infer that the hot CGM observed via X-ray emission has its origins in both hierarchical accretion and stellar recycling in that the majority of L_* galaxies develop quasi-hydrostatic coronae through shock heating and adiabatic compression of gas accreted from the IGM, supplemented by relatively small amounts of gas recycled through the galaxy by stellar feedback. They conclude that the hot corona is primarily primordial gas and is forged via accretion during galaxy assembly.

Though the range of hot coronal metallicities determined from X-ray luminosities might suggest near solar enrichment, Crain et al. (2013) find that luminosity-weighting of X-ray measurements bias the perceived metallicity of hot coronal gas. In their simulations, L_X weighted metallicities are Z_gas ∼ 1, but gas mass-weighted metallicities are Z_gas ∼ 0.1 (with a very shallow trend for metallicity to decrease with increasing virial mass).

Hodges-Kluck & Bregman (2013) reported Z_gas ∼ 0.1 for NGC 891 (a late type galaxy), and argue that the primary source of the gas is IGM accretion. Even at higher redshifts than covered by our sample of galaxies, simulations suggest that the hot coronal metallicity is consistent with Z_gas ∼ 0.1. Shen et al. (2012), using their ErisMC simulations, find total gas metallicities of Z_gas ≃ 0.08 at ≃ 100 kpc at z = 3, consistent with recent observations of circumgalactic metals around Lyman break galaxies. In the owls simulations, van de Voort & Schaye (2012) find Z_gas ∼ 0.1 in the "hot mode" phase of the CGM (T_max > 10^5.5 K) for the virial mass range log M_h/M_☉ = 10–13 at z = 2.

Based upon the above considerations, we adopt Z_gas = 0.1 for our fiducial model for computing the cooling radius for our galaxy sample. In Figure 11(c), we plot the redshift evolution of R_c for Z_gas = 0.1. Shown are z = 0, 0.3, 0.5, 0.7, and 0.9. At fixed mass and metallicity, evolution in the cooling radius is dominated by the formation time of the halo and the concentration parameter, which sets the hot gas density scale via g[C_v(M_h, z_gal)], the scale radius via R_s = R_vir/C_v, and the gas temperature via $V_c^{\rm max}$, since R_max is proportional to R_s.