AN EMPIRICAL CHARACTERIZATION OF EXTENDED COOL GAS AROUND GALAXIES USING Mg ii ABSORPTION FEATURES^*

We have obtained optical spectra of 89 photometrically selected SDSS galaxies, using the Double Imaging Spectrograph (DIS; Lupton 1995) on the 3.5 m telescope at the Apache Point Observatory and MagE (Marshall et al. 2008) on the Magellan Clay Telescope at the Las Campanas Observatory. All of these galaxies satisfy the selection criteria described in Section 2. The medium-to-high-resolution spectra allow us to obtain precise and accurate redshift measurements of these galaxies, which are necessary for establishing (or otherwise) a physical connection with absorbers found along nearby QSO sightlines. The spectroscopic observations of these SDSS galaxies are summarized here.

Long-slit spectra of 41 galaxies were obtained using DIS over the period from 2008 October through 2009 June. The blue and red cameras have a pixel scale of 04 and 042 per pixel, respectively, on the 3.5 m telescope. We used a 15 slit, the B400 grating in the blue channel for a dispersion of 1.83 Å per pixel, and the R300 grating in the red channel for a dispersion of 2.31 Å per pixel. The blue and red channels with the medium-resolution gratings together offer contiguous spectral coverage from λ = 3800 Å to λ = 9800 Å and a spectral resolution of FWHM ≈ 500 km s⁻¹ at λ = 4400 Å and FWHM ≈ 400 km s⁻¹ at λ = 7500 Å. The observations were carried out in a series of two to three exposures of between 1200 s and 1800 s each, and no dither was applied between individual exposures. Flat-field frames were taken in the afternoon prior to the beginning of each night. Calibration frames for wavelength solutions were taken immediately after each set of science exposures using the truss lamps on the secondary cage. When the sky was clear, we also observed a spectrophotometric standard star observed at the beginning of each night for relative flux calibration.

All of the DIS spectroscopic data were reduced using standard long-slit spectral reduction procedures. The spectra were calibrated to vacuum wavelengths, corrected for the heliocentric motion, and flux-calibrated using a sensitivity function derived from observations of a flux standard. Redshifts of the galaxies were determined based on a cross-correlation analysis with a linear combination of SDSS eigen spectra of galaxies. The typical redshift uncertainty was found to be Δz ∼ 0.0003. An example of the DIS galaxy spectra is presented in the top panel of Figure 1.

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Optical spectra of 48 galaxies were obtained using MagE in 2008 September, 2009 February, 2009 March, and 2009 June. The camera has a plate scale of 03 per pixel. We used a 1″ slit and 2 × 1 binning during readout, which yielded a spectral resolution of FWHM ≈ 150 km s⁻¹. Depending on the brightness of the galaxies, the observations were carried out in a sequence of one to two exposures of duration 300–1200 s each. The galaxy data were processed and reduced using data reduction software that was originally developed by G. Becker and later modified by us to work with binned spectral frames. Wavelengths were calibrated using a ThAr frame obtained immediately after each exposure and subsequently corrected to vacuum and heliocentric wavelengths. Flux calibration was performed using a sensitivity function derived from observations of the flux standard GD50. Individual flux-calibrated orders were co-added to form a single spectrum that covers a spectral range from λ ≈ 4000 Å to λ = 9600 Å. Redshifts of the galaxies were determined based on a cross-correlation analysis with a linear combination of SDSS eigen spectra of galaxies. The typical redshift uncertainty was found to be Δz ∼ 0.0001. An example of the MagE galaxy spectra is presented in the bottom panel of Figure 1.

We have also located five additional galaxies, which satisfy the selection criteria described in Section 2 and already have optical spectra and robust spectroscopic redshifts available in the public SDSS data archive. These galaxies are included in our final galaxy sample for the search of Mg ii absorbers. A journal of the spectroscopic observations of the full galaxy sample is in Table 1.

Echellette spectroscopic observations of 70 QSOs were obtained using the MagE spectrograph (Marshall et al. 2008) on the Magellan Clay Telescope over the period from 2008 January through 2009 June. We used a 1″ slit for the majority of the QSOs (with a few sources taken with a 07 slit) and 1 × 1 binning during readout, which yielded a spectral resolution of FWHM ≈ 70 km s⁻¹ (or FWHM ≈ 50 km s⁻¹ for spectra taken with a 07 slit). Depending on the brightness of the QSOs, the observations were carried out in a sequence of one to three exposures of duration 600–2400 s each. The QSO spectra were processed and reduced using a reduction pipeline developed by G. Becker and kindly offered to us by the author. Wavelengths were calibrated using a ThAr frame obtained immediately after each exposure and subsequently corrected to vacuum and heliocentric wavelengths. Flux calibration was performed using a sensitivity function derived from observations of the flux standard GD50. Individual flux-calibrated echellette orders were co-added to form a single spectrum that covers a spectral range from λ = 3050 Å to λ = 1 μm. These order-combined individual exposures were then continuum normalized and stacked to form one final combined spectrum per QSO using an optimal weighting routine. The continuum was determined using a low-order polynomial fit to spectral regions that are free of strong absorption features. The flux-calibrated spectra have S/N ≳10 per resolution element across the entire spectral range. Examples of an order-combined spectrum and a continuum-normalized final stack are shown in Figure 2. A journal of the MagE QSO observations is in Table 2.

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We have established a sample of 94 galaxies with r′ < 22.3 and robust spectroscopic redshifts at z < 0.8 in fields around 70 distant background QSOs (z_QSO > 0.6). These 94 spectroscopically confirmed galaxies form the “full” sample of our study. Within the full sample, 17 galaxies are found with at least one close neighbor at $\rho <{\hat{R}}_{\rm gas}$ and velocity separation Δ v < 300 km s⁻¹, including one galaxy that occurs in the vicinity of a background QSO at z ≈ 0.8. The presence of a close neighbor implies that the galaxy is likely to reside in a group environment. The association between absorbers and individual galaxies becomes uncertain in a group environment. In addition, interactions between group members are expected to alter the properties of gaseous halos (e.g., Gunn & Gott 1972; Balogh et al. 2000; Verdes-Montenegro et al. 2001). Therefore, these galaxies form a “group”-galaxy subsample and are considered separately from the remaining “isolated”-galaxy subsample in our analysis below. Here, we summarize the galaxy properties in the full sample.

We first examine the redshift distribution of the galaxies. Figure 3 shows in the left panel the redshift distribution of all 94 spectroscopically confirmed galaxies in our survey (full histogram) and the redshift distribution of the “isolated”-galaxy subsample (shaded histogram). Figure 4 shows the comparison of photometric redshifts z_phot and spectroscopic redshift z_spec for galaxies in our full sample. Excluding the single outlier at z ≈ 0.8, we find that SDSS photometric redshifts for galaxies brighter than r′ = 21.6 are accurate to within a mean residual of 〈(z_phot − z_spec)/(1 + z_spec) 〉 = 0.01 and a corresponding rms scatter of δ z ≡ Δ z/(1 + z) = 0.06. There exists an increased uncertainty in the photometric redshifts for galaxies at z ≳ 0.35, resulting in the apparent “tilt” seen in the left panel. We also examine photometric redshift uncertainties versus galaxy brightness and intrinsic color. The right panel of Figure 4 shows that the precision and accuracy of photometric redshifts are still higher for brighter (r′ < 20) and redder galaxies (with rest-frame B_AB − R_AB color consistent with elliptical/S0 galaxies; see Equation (4) and Figure 5 for the classification of different galaxy types). With the exception of one outlier at z ≈ 0.8, the target selection based on known photometric redshifts has allowed us to effectively identify foreground galaxies at z = 0.1–0.4 that occur at close projected distances to the line of sight toward a background QSO.

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Next, we present the impact parameter distribution of the galaxies to a QSO sightline in the right panel of Figure 3. “Group” galaxies are displayed in open symbols and “isolated” galaxies in solid symbols. Galaxies in different luminosity ranges are shown in different shapes of symbols. Figure 3 shows that luminous galaxies with B-band luminosity $L_B>L_{B_*}$ (circles)⁶ span a range in their projected distance to a QSO sightline from ρ = 15.2 h⁻¹ kpc to ρ = 119.2 h⁻¹ kpc, while sub-L_* galaxies of $L_B=(0.1-1)\,L_{B_*}$ (triangles) cover a range from ρ = 8.5 h⁻¹ kpc to ρ = 70.3 h⁻¹ kpc. Our sample also includes two faint dwarfs with $L_B<0.1\,L_{B_*}$ (squares) at ρ < 15 h⁻¹ kpc. In addition, the QSO point spread function prevents us from finding galaxies at angular distances Δ θ ≲ 3″ from the QSO sightlines in the SDSS catalog. The minimum project distances we can probe using the SDSS galaxy sample changes from ρ_min < 10 h⁻¹ kpc at z ≲ 0.15 to ρ_min > 15 h⁻¹ kpc at z ≳ 0.4. The distribution shows that our survey is designed to study halo gas (rather than the ISM) of distant galaxies. Our sample includes more luminous galaxies (presumably more massive and therefore residing in more extended gaseous halos) at larger impact parameters with the maximum survey radius determined by the fiducial model (Section 2).

The rest-frame absolute B-band magnitudes and rest-frame B−R colors of all 94 galaxies are calculated based on the spectroscopic redshifts. To carry out this calculation, we first adopt the best-fit model magnitudes from the SDSS photometric catalog and compute the composite model magnitudes in the u′g′r′i′z′ bandpasses following the instructions of Scranton et al. (2005).⁷ Then, the composite model magnitudes are corrected for the Galactic extinctions following Schlegel et al. (1998). Finally for each galaxy, we identify the bandpasses that correspond most closely to the rest-frame B and R bands based on its spectroscopic redshift. We interpolate between the composite model magnitudes to determine the intrinsic B−R color and the k-term necessary to correct the remaining color difference for deriving the rest-frame absolute B-band magnitude following Equation (3).

Figure 5 shows the redshift distribution of rest-frame absolute B-band magnitudes of our galaxies, spanning a range from M_B − 5log  h = −16.22 to M_B − 5log  h = −22.5. We also examine the underlying stellar population by comparing the observed optical colors with expectations from known spectral energy distribution templates of galaxies of different morphological type. We consider the E/S0, Sbc, Scd, and Irr galaxy templates from Coleman et al. (1980) and calculate the expected colors versus redshift under a no-evolution assumption (see also Fukugita et al. 1995). The right panel of Figure 5 shows that the optical colors of our galaxies are in broad agreement with the expected colors of galaxies across a broad range of morphology. We classify the galaxies based on the intrinsic B_AB − R_AB color according to the following criteria:

$$\begin{eqnarray} \begin{array}{ll} B_{AB}-R_{AB} > 1.1 & {\rm elliptical/S0,} \\ 0.6 < B_{AB}-R_{AB} \le 1.1 & {\rm disk\; galaxies,} \\ B_{AB}-R_{AB} < 0.6 & {\rm irregular/star}\hbox{-}{\rm forming\ galaxies.} \end{array} \end{eqnarray} \tag{ 4 }$$

Of the 94 galaxies in our spectroscopic sample, 18 have the intrinsic B_AB − R_AB color consistent with an elliptical or S0 galaxy, 52 are consistent with a disk galaxy, and 24 are consistent with an irregular or younger galaxies. The classification is qualitatively consistent with the spectral features seen in the galaxy spectra. Of the 18 galaxies classified as elliptical or S0, more than 2/3 display spectral features dominated by absorption transitions due to Ca ii H and K and G band. A more detailed analysis is presented in a separate paper (J. E. Helsby et al. 2010, in preparation).

To establish a galaxy–Mg ii absorber pair sample for the subsequent analysis, we examine the echellette spectra of the background QSOs and search for the corresponding Mg ii absorption doublet at the locations of the galaxies presented in Section 4.1. Specifically, we first search for the corresponding doublet features within velocity separation Δv = ±1000 km s⁻¹ of the galaxy redshifts. Then we accept absorption lines according to a 2σ detection threshold criterion, which is appropriate because the measurements are performed at known galaxy redshifts. Next, we determine the absorber redshift based on the best-fit line centroid of a Gaussian profile analysis of the doublets. Uncertainties in the absorber redshifts are less than δv = (c Δ z/(1 + z) = 25 km s⁻¹. Next, we measure 2σ upper limits to the 2796 Å absorption equivalent widths for galaxies that are not paired with corresponding absorbers.

The procedure identifies 47 Mg ii absorbers and 24 upper limits in the vicinities of 71 isolated galaxies, and seven Mg ii absorbers and one upper limit around 17 “group” galaxies. We are unable to obtain significant constraints for the presence/absence of Mg ii absorbers around five galaxies, because the expected Mg ii absorption features are blended with either strong C iv λλ 1548, 1550 absorption features at higher redshifts or the atmosphere O 3 absorption complex at λ ≈ 3200. Finally, one galaxy is found at z_spec = 0.0934, outside of the redshift range targeted for the MagE Mg ii absorber survey. Examples of the galaxy–Mg ii absorber pairs, including upper limits, are presented in Figure 6. A summary of the properties of the spectroscopically confirmed galaxies is presented in Table 3, where we list for each galaxy in Columns 1–8 the ID, right ascension, and declination offsets from the QSO Δα and Δδ, redshift z_gal, impact parameter ρ, apparent r′-band magnitude, B_AB − R_AB color, and absolute B-band magnitude M_B − 5log h. Measurement uncertainties in r′ and M_B − 5log h are typically 0.05 dex.

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Impact parameter separations of the galaxy and absorber pairs in the “isolated” sample range from ρ = 6.1 to 119.2 h⁻¹ physical kpc with a median of 〈ρ〉_med = 27.9 h⁻¹ kpc. The redshifts of the “isolated” galaxies range from z = 0.1097 to z = 0.4933 with a median of 〈z〉_med = 0.2533. The rest-frame absolute B-band magnitudes of the galaxies span a range from M_B − 5log  h = −16.4 to M_B − 5log  h = −21.4 with a median of 〈M_B − 5log  h〉 = −19.6. The rest-frame B_AB − R_AB colors range from B_AB − R_AB ≈ 0 to B_AB − R_AB ≈ 1.5 with a median of 〈B_AB − R_AB〉_med ≈ 0.8.

The redshift and absorption equivalent width of the corresponding Mg ii absorption feature for each of the 71 galaxies in our sample are listed in Columns 9 and 10 of Table 3. Figure 7 shows the relative velocity distribution of the 47 galaxy–Mg ii absorber pairs in the isolated sample. We find that with the exception of one pair, which exhibits a velocity separation of $\,v_{\rm Mg\,\mathsc {ii}-Galaxy}=-645$ km s⁻¹, the velocity separation between the Mg ii absorbers and their member galaxies is well characterized by a Gaussian distribution of mean velocity difference $\langle \,v_{\rm Mg\,\mathsc {ii}-Galaxy}\rangle =16$ km s⁻¹ and dispersion σ = 137 km s⁻¹. The velocity dispersion between the galaxy–Mg ii absorber pair sample is comparable to the velocity dispersion seen in a Milky-Way size galaxy (see also Steidel et al. 2002; Chen et al. 2005; Kacprzak et al. 2010 for rotation curve comparisons of galaxies and strong absorbers at intermediate redshifts), supporting a physical association between the absorbing gas and the member galaxy.

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To obtain the best-fit model that characterizes the correlation between the strength of Mg ii absorbers and the properties of galaxies, we adopt a parameterized functional form

$$\begin{equation} y = f(x1,x2,\ldots), \end{equation} \tag{ 5 }$$

where y is the predicted absorber strength and is always log  W_r(2796) in our analysis, and x_i's are independent measurements of galaxy properties such as galaxy impact parameter ρ, rest-frame absolute magnitude M_B − 5 log  h, redshift z, or intrinsic color B_AB − R_AB. We consider a simple power-law model for characterizing the W_r(2796) versus ρ relation. In logarithmic space, the model is expressed as a linear function according to

$$\begin{equation} \log \bar{W}_r^{\rm p}(2796)\,=\,a_0 + a_1 \log \,\rho + a_2 (M_B-M_{B_*}) +\cdots{.} \end{equation} \tag{ 6 }$$

To determine the values of different coefficients a_i's, we perform a maximum-likelihood analysis that includes the upper limits in our galaxy and Mg ii absorber pair sample. The likelihood function of this analysis is defined as

$$\begin{eqnarray} {\cal L}(\bar{y}) & = &\left(\prod _{i=1}^{n} \exp \left\lbrace -\frac{1}{2} \left[ \frac{y_i - \bar{y}(\rho _i, L_{B_i})}{\sigma _i} \right]^2 \right\rbrace \right) \nonumber \\ & & \times \left(\prod _{i=1}^m \int _{y_i}^{-\infty } dy^{\prime } \exp \left\lbrace -\frac{1}{2} \left[ \frac{y^{\prime } - \bar{y}(\rho _{i}, L_{B_i})}{\sigma _i} \right]^2 \right\rbrace \right){,}\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where y_i = log  W_i is the observed value of galaxy i, $\bar{y}=\log \,\bar{W}_r(2796)$ is the model expectation, and σ_i is the measurement uncertainty of y_i. The first product of Equation (7) extends over the n measurements and the second product extends over the m upper limits. (This definition of the likelihood function is appropriate if the residuals about the mean relationship are normally distributed.) We also consider the possibility that σ_i includes a significant intrinsic scatter (which presumably arises due to intrinsic variations between individual galaxies) as well as measurement error. We express σ_i as a quadratic sum of the cosmic scatter σ_c and the measurement error $\sigma _{m_i}$

$$\begin{equation} \sigma _i^2 = \sigma _c^2 + \sigma _{m_i}^2, \end{equation} \tag{ 8 }$$

where the intrinsic scatter is defined by

$${\selectfont\fontsize{8}{9}{\begin{equation} \sigma _c^2 = {\rm med} \left(\left\lbrace y_i - \bar{y}(\rho _{i},L_{B_i},...) - \frac{1}{n} \sum _{j=1}^n [ y_j - \bar{y}(\rho _{j},L_{B_j},...) ] \right\rbrace ^2 - \sigma _{m_i}^2 \right). \end{equation}}} \tag{ 9 }$$

Because σ_c depends on the maximum-likelihood solution $\bar{y}=\log \,\bar{W}(\rho _{i},L_{B_i})$, the maximum-likelihood solution is obtained iteratively with respect to Equations (7) and (9).

Adopting a power-law function for describing the W_r(2796) versus ρ distribution in Figure 8, we find based on the maximum-likelihood analysis a best-fit model of

$$\begin{equation} \log \bar{W}_r^{\rm p}(2796) = -(1.17\pm 0.10) \log \rho + (1.28\pm 0.13) \end{equation} \tag{ 10 }$$

with an intrinsic scatter of σ_c = 0.195. The rms residual between the observed and model Mg ii absorber strengths is found to be ${\rm rms}(\log \,W_r-\log \,\bar{W})=0.350$. Errors associated with the best-fit coefficients are 1σ uncertainties. We present the best-fit model and the intrinsic scatter in Figure 8.

The results of the likelihood analysis demonstrate at a high confidence level that the Mg ii absorber strength W_r(2796) scales inversely with galaxy impact parameter. To assess the significance of this anti-correlation, we perform a generalized Kendall test (Feigelson & Nelson 1985) that accounts for the presence of non-detections and find that the observed distribution of W_r(2796) versus ρ deviates from a random distribution at more than 5.5σ level of significance.

Despite a statistically significant anti-correlation seen between W_r(2796) and ρ, there exists a large scatter between the data and the best-fit power-law model (Equation (10)). Previous authors (Chen & Tinker 2008; Kacprzak et al. 2008) have shown that the observed Mg ii absorber strength scales with galaxy B-band luminosity. Recall that our galaxy sample spans a range in rest-frame absolute B-band magnitude from M_B − 5log  h = −16.4 to M_B − 5log  h = −21.4. Here, we examine whether including intrinsic B-band luminosity helps to further strengthen this anti-correlation.

Adopting a power-law function to parameterize the scaling relation, we find based on the likelihood analysis that the observed absorber strength is best described by $\log W_r(2796) = -(1.52\pm 0.10) \log \rho - (0.20\pm 0.03)\,(M_B-M_{B_*}) + (1.88\pm 0.15)$ (dotted line in Figure 9), with an intrinsic scatter of σ_c = 0.175 and ${\rm rms}(\log \,W_r(2796)-\log \,\hat{W})=0.318$. We adopt $M_{B_*}-5\,\log \,h=-19.8$ from Faber et al. (2007), which best characterizes the blue galaxy population at z ∼ 0.4. The result of the likelihood analysis shows that including the scaling with B-band luminosity indeed reduces the intrinsic scatter by ≈12%. At the same time, five outliers at >3 σ_c from the mean relation (points in dotted circles) have now become apparent after accounting for the scaling with B-band luminosity.

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Inspecting the imaging and spectral properties of the outliers, we note that the discrepant triangle and circle in the lower-center part of the graph are, respectively, an active galaxy showing broad emission lines and a red, evolved galaxy with a possible companion blended with a nearby bright star. The discrepant triangle in the upper left is the galaxy–Mg ii absorber pair with a large velocity separation of Δ v ≈ −645 km s⁻¹ in Figure 7. The discrepant triangle and circle in the upper right are galaxies with likely companions at close distances. Follow-up spectroscopy of these surrounding galaxies will confirm the nature of these close neighbors.

Excluding the five outliers, we obtain a revised best-fit power-law model of

$$\begin{eqnarray} \log \bar{W}_r^{\rm p}(2796) &=& {-}(1.93\pm 0.11) \log \rho - (0.27\pm 0.02)\nonumber\\ &&\times (M_B-M_{B_*}) + (2.51\pm 0.16), \end{eqnarray} \tag{ 11 }$$

with an intrinsic scatter of σ_c = 0.196 and ${\rm rms}(\log W_r-\log \,\bar{W})=0.235$. We find that after accounting for an optimal scaling relation with L_B, the slope of the anti-correlation is steepened. We present the revised best-fit model as solid line in Figure 9. A generalized Kendall test demonstrates that the distribution of W_r(2796) versus ρ after accounting for galaxy luminosity deviates from a random distribution at more than 7σ level of significance. It indicates that the probability that W_r(2796) is randomly distributed with respect to ρ is ≪0.1%.

Here, we examine whether including additional galaxy properties, such as B_AB − R_AB color and redshift, helps to further strengthen this anti-correlation. Recall that our galaxy sample spans a range in B_AB − R_AB color from B_AB − R_AB ≈ 0 to B_AB − R_AB ≈ 1.5, and a range in redshift from z = 0.1097 to z = 0.4933.

Including the intrinsic B_AB − R_AB color of the galaxies as an additional independent variable in the power-law model and excluding the outliers in Figure 9, we find based on the likelihood analysis (Equations (7)–(9)) that

$$\begin{eqnarray} \log \bar{W}_r^{\rm p}(2796) &=& {-}(1.92\pm 0.11) \log \rho - (0.27\pm 0.02) \nonumber \\ &&\times (M_B-M_{B_*})+ (0.05\pm 0.02)\nonumber\\ &&\times (B_{AB}-R_{AB})+ (2.46\pm 0.15), \end{eqnarray} \tag{ 12 }$$

with an intrinsic scatter of σ_c = 0.184 and ${\rm rms}(\log \,W_r-\log \,\bar{W})=0.246$. The small scaling coefficient of the color term in Equation (12) indicates that the observed W_r(2796) versus ρ anti-correlation depends only weakly on galaxy intrinsic colors.

Next, we examine whether including galaxy redshift as an additional independent variable in the power-law model improves the strength of the anti-correlation. Excluding the outliers, we obtain a best-fit model of

$$\begin{eqnarray} \log \bar{W}_r(2796) &=& {-}(1.90\pm 0.11) \log \rho - (0.25\pm 0.03) \nonumber \\ && \times(M_B-M_{B_*})+ (0.73\pm 0.50)\,\log \frac{(1+z)}{(1+z_0)} \nonumber\\ && + (2.47\pm 0.15), \end{eqnarray} \tag{ 13 }$$

where z₀ = 0.25 which is the median redshift of the “isolated”-galaxy sample. The best-fit relation has an intrinsic scatter of σ_c = 0.184 and ${\rm rms}(\log \,W_r-\log \,\bar{W})=0.245$. The uncertain scaling coefficient of the redshift term in Equation (13) indicates that there is little evolution in the extended Mg ii halos around galaxies between z ≈ 0.1 and z ≈ 0.5.

The analysis presented in Section 5.2 shows that the strengths of Mg ii absorbers depend on the projected distances and intrinsic luminosities of the absorbing galaxies. In addition to the best-fit scaling relation, we also note that with the exception of two outliers all 33 galaxies at luminosity-scaled impact parameter $\rho ^{\prime }\equiv \rho \times \,(L_B/L_{B_*})^{-0.35}<30\ h^{-1}$ (Equation (11)) have a coincident Mg ii absorber of W_r(2796)>0.3 Å. The observed high incidence of strong Mg ii absorbers around galaxies with a broad range of intrinsic colors strongly supports the notion that extended Mg ii halos are a common and generic feature of galaxies of all types, from evolved early-type galaxies to late-type star-forming systems.

While extended Mg ii absorbing gas reaches out to larger projected distances, our sample shows that both the absorption strength and gas covering fraction declines toward larger radii. Here, we examine how the gas covering fraction κ varies with ρ and W_r(2796).

We perform a maximum-likelihood analysis to determine κ. The probability that a galaxy gives rise to an absorption system of some absorption equivalent width threshold is written as

$$\begin{equation} P(\kappa)=\kappa (\rho,W_0) \, B [r_1(L_B),r_2(L_B);\rho ], \end{equation} \tag{ 14 }$$

where κ is the fraction of galaxies that give rise to Mg ii absorption, and B is a boxcar function that defines the impact parameter interval; B = 1 if r₁(L_B) ⩽ ρ < r₂(L_B) and B = 0 otherwise. Equation (14) takes into account the scaling relation between the gaseous extent of galaxies and galaxy B-band luminosity. The likelihood of detecting an ensemble of galaxies, n of which give rise to Mg ii absorption systems and m of which do not, is given by

$$\begin{eqnarray} {\cal L}(\kappa)&=&\prod _{i=1}^n\,\kappa (\rho _i,W_i) B\big[r_1\big(L_{B_i}\big),r_2\big(L_{B_i}\big);\rho _i\big]\nonumber\\ &&\times \prod _{j=1}^m\,\big\lbrace 1-\kappa (\rho _j,W_j) B\big[r_1\big(L_{B_j}\big),r_2\big(L_{B_j}\big);\rho _j\big]\big\rbrace \nonumber \\ &=& \prod _{i=1}^n\,\langle \kappa \rangle \,\prod _{j=1}^m(1-\langle \kappa \rangle) \nonumber \\ &=& \langle \kappa \rangle ^n\,(1-\langle \kappa \rangle)^m. \end{eqnarray} \tag{ 15 }$$

We evaluate κ for three absorption equivalent width thresholds, W₀ = 0.5 Å, W₀ = 0.3 Å, and W₀ = 0.1 Å. In the “isolated” galaxy sample, all MagE spectra of the corresponding QSOs have sufficient S/N for detecting an absorber of W_r(2796) ⩾ 0.3 Å and 61 have sufficient S/N spectra for uncovering an absorber of W_r(2796) ⩾ 0.1 Å. We calculate κ for different impact parameter intervals, using the best-fit scaling relation of Equation (11) and excluding the outliers. The results are shown in Figure 10. The error bars indicate the 68% confidence interval of the estimated κ(ρ, W₀). The impact parameter intervals are chosen to contain ten galaxy–absorber pairs per bin, except for the last bin that includes only the remaining seven pairs at largest separations. Figure 10 shows that the covering fraction of W_r(2796) ⩾ 0.1 Å absorbers varies from 100% within ρ = 30 h⁻¹ kpc of an L_* galaxy to <20% beyond ρ = 60 h⁻¹ kpc.

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The large gas covering fraction seen in our galaxy sample at small projected distances is in stark contrast to the finding of Gauthier et al. (2010), who have compared the incidence of Mg ii absorbers around LRGs of i′ < 20 mag at z ≈ 0.5. These authors have found that the covering fraction of extended Mg ii absorbers of W_r(2796) ⩾ 0.5 Å is no more than 40% at ρ < 50 h⁻¹ kpc (or ρ′ < 35 h⁻¹ kpc after accounting for the luminosity scaling) and <30% at ρ < 100 h⁻¹ kpc (or ρ′ < 70 h⁻¹ kpc) from LRGs. At W_r(2796) ⩾ 0.5, our galaxy sample shows κ_0.5 ≈ 83% at ρ′ < 30 h⁻¹ kpc.

We note that the LRGs are luminous with M_B − 5 log  h < −21.35 and are understood to reside in massive halos of M_h ≳ 10¹³h^-1 M_☉, whereas the galaxies in our sample are fainter, with a median rest-frame B-band magnitude of 〈M_B − 5log  h〉 = −19.6, and presumably reside in lower-mass halos of M_h ∼ 10^12.3h^-1 M_☉ at z ≈ 0.25. To examine how the covering fraction of Mg ii absorbing gas varies with galaxy luminosity, we first divide our galaxy sample into luminous ($L_B>L_{B_*}$) and faint ($L_B\le L_{B_*}$) subsamples and then determine the mean gas covering fraction $\langle \kappa _{W_0}\rangle$ over the entire gaseous halo defined by R_gas (see Section 6.2) at a given W_r(2796) threshold, W₀.

We consider three different threshold values, W₀ = 0.1 Å, W₀ = 0.3 Å, and W₀ = 0.5 Å. The results are presented in Figure 11, together with the measurement of Gauthier et al. (2010) for W₀ = 0.5 Å around LRGs. The decreasing mean covering fraction over a fixed halo radius with increasing absorber strength is consistent with the expectation from a decreasing cloud density profile toward large radii (see Section 6.2). In addition, we find little dependence between the mean gas covering fraction and galaxy luminosity within our sample, but a significant reduction in gas covering fraction around LRGs. The declining gas covering fraction of strong Mg ii absorbers with increasing galaxy luminosity (halo mass) is qualitatively consistent with the expectation from the observed clustering properties of Mg ii absorbers (Tinker & Chen 2008, 2009) and the theoretical expectation of diminishing cool gas fraction in massive halos (Kereš et al. 2005, 2009; Dekel & Birnboim 2006).

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The best-fit power-law model presented in Equation (11) and Figure 9 indicates that the observed absorber strength W_r(2796) scales inversely with increasing projected distance ρ with a steep slope of Δlog  W_r(2796)/Δlog  ρ = −1.9. While the power-law model appears to describe the observations well, a physical interpretation of this steep anti-correlation is not straightforward. Using a small sample of 13 galaxy–Mg ii absorber pairs and 10 galaxies at ρ < 100 h⁻¹ kpc that do not give rise to Mg ii absorption to a sensitive upper limit (W_r(2796) ≲ 0.02 Å), Chen & Tinker (2008) showed that the anti-correlation between W_r(2796) and ρ, after accounting for B-band luminosity distribution, is well described by either an isothermal density profile or a Navarro–Frenk–White (NFW; Navarro et al. 1996) profile of the absorbing clumps with a finite extent R_gas. An isothermal density profile is motivated by the observed rotation curves of nearby galaxies, while an NFW profile is found to represent the density profiles of dark matter halos in high-resolution numerical simulations. Using the new sample of galaxy–Mg ii absorber pairs that is three times of the Chen & Tinker (2008) sample, we examine whether the anti-correlation displayed in Figure 9 is better described by either an isothermal density profile or an NFW profile.

For an isothermal profile of gaseous clumps within a finite extent R_gas, the W_r(2796) versus ρ relation is characterized following Chen & Tinker (2008) as

$$\begin{equation} \bar{W}_r^{\rm iso}(2796) =\frac{W_0}{\sqrt{\rho ^2\big/a_h^2+1}}\tan ^{-1}{\sqrt{\frac{R_{\rm gas}^2-\rho ^2}{\rho ^2+a_h^2}}} \end{equation} \tag{ 16 }$$

at ρ ⩽ R_gas and W_r(2796) = 0 otherwise. The core radius a_h is defined to be a_h = 0.2 R_gas and does not affect the expected W_r(2796) at large ρ. The extent of Mg ii absorbing gas scales with the luminosity of the absorbing galaxy according to

$$\begin{equation} \frac{R_{\rm gas}}{R_{{\rm gas}*}}=\left(\frac{L_B}{L_{B_*}}\right)^{\beta }. \end{equation} \tag{ 17 }$$

Following the expectations of an isothermal model in Equations (16) and (17), we perform the likelihood analysis described in Equations (7)–(9) to find the best-fit values of W₀, $R_{\rm gas_*}$, and β. Excluding the outliers, the results of the likelihood analysis show that the observations are best described by log  W^iso₀ = 1.24 ± 0.03,

$$\begin{equation} \beta ^{\rm iso}=0.35_{-0.04}^{+0.01}, \end{equation} \tag{ 18 }$$

and

$$\begin{equation} \log \,R_{\rm gas_*}^{\rm iso}=1.87\pm 0.01. \end{equation} \tag{ 19 }$$

The errors indicate the 95% confidence intervals. The best-fit isothermal profile is also characterized by an intrinsic scatter of σ_c = 0.104 and an rms residual between the observed and model Mg ii absorber strengths of ${\rm rms}(\log W_r-\log \bar{W})=0.233$.

For an NFW profile, the absorber strength is expected to vary with ρ according to

$$\begin{equation} \bar{W}_r^{\rm NFW}(2796)=\displaystyle \int _0^{\sqrt{R_{\rm gas}^2-\rho ^2}}\frac{W_0\,r_s^3}{(r_s+\sqrt{l^2+\rho ^2})^2\sqrt{l^2+\rho ^2}}\,dl, \end{equation} \tag{ 20 }$$

where r_s is the scale radius. We adopt r_s ≡ R₂₀₀/15 that gives a halo concentration index of 15 (e.g., Dolag et al. 2004). We perform the likelihood analysis described in Equations (7)–(9) and find log  W^NFW₀ = −0.56 ± 0.05,

$$\begin{equation} \beta ^{\rm NFW}=0.35\pm 0.03, \end{equation} \tag{ 21 }$$

and

$$\begin{equation} \log R_{\rm gas_*}^{\rm NFW}=1.89_{-0.12}^{+0.05}. \end{equation} \tag{ 22 }$$

The errors indicate the 95% confidence intervals. The best-fit NFW profile has associated intrinsic scatter of σ_c = 0.175 and rms residual between the observed and model Mg ii absorber strengths of ${\rm rms}(\log \,W_r-\log \,\bar{W})=0.253$.

Figure 12 displays the best-fit models in comparison to observations. While the isothermal model provides a somewhat better characterization of the observations in terms of a minimum derived intrinsic scatter and rms residual, both models result in a consistent best-fit scaling relation of

$$\begin{equation} \frac{R_{\rm gas}}{R_{\rm gas_*}}= \left(\frac{L_B}{L_{B_*}}\right)^{0.35}, \end{equation} \tag{ 23 }$$

where the characteristic gaseous radius of an L_* galaxy is found to be

$$\begin{equation} R_{\rm gas_*}\approx 75\ h^{-1}\,{\rm kpc} \end{equation} \tag{ 24 }$$

at z ∼ 0.25.

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It is clear that the scatter between the observations and the best-fit isothermal model after accounting for the scaling relation with galaxy luminosity remains large in Figure 12. A natural expectation of the isothermal model in Equation (16) is that W₀ depends on galaxy properties, particularly on L_B or mass, because more luminous (and therefore more massive) galaxies possess more extended gaseous halos and would exhibit stronger W₀ (e.g., Tinker & Chen 2010). However, we find no correlation between the residuals of $(\log \,W_r-\log \,\bar{W})$ in Figure 12 and absolute B-band magnitude of the galaxies, indicating that W₀ does not depend strongly on galaxy luminosity/mass. The lack of correlation suggests that the mean absorber strength per halo does not vary with halo mass and that the cool gas fraction declines with increasing halo mass in the mass regime probed by our sample.

The best-fit scaling power in Equation (23) confirms the earlier result of Chen & Tinker (2008). The best-fit characteristic gaseous radius shown in Equation (24) is smaller than the earlier estimate of 91⁺³₋₈ h⁻¹ kpc from a smaller galaxy sample, but they are consistent to within 2σ error uncertainties. The new galaxy sample presented in this paper, which is three times larger than the initial sample employed by Chen & Tinker, has revealed a substantial scatter, particularly at large radii. The large uncertainty in $R_{\rm gas_*}$ can be understood as due to the inherent degeneracy between the spatial extent R_gas and diminishing covering fraction κ(ρ) of Mg ii absorbing gas at large radii. A larger $R_{\rm gas_*}$ can be accommodated with a smaller gas covering fraction. For example, within $R_{\rm gas_*}=75\ h^{-1}$ kpc, 39 of the 59 galaxies have an associated Mg ii absorber of W_r(2796) ⩾ 0.3 Å. This yields a gas covering fraction of κ_0.3 ≈ 66% at ρ′ < 75 h⁻¹ kpc. Within $R_{\rm gas_*}=91\ h^{-1}$ kpc, we find κ_0.3 ≈ 61% at ρ′ < 91 h⁻¹ kpc after excluding outliers.

In Chen & Tinker (2008), the best-fit $R_{\rm gas_*}=91\ h^{-1}$ kpc was close to the maximum impact separation of r₀ = 100 h⁻¹ kpc in the galaxy–QSO pair sample. The selection criterion was imposed by these authors to minimize the incidence of correlated galaxies through large-scale clustering (see their Section 2 for discussion). Chen & Tinker examined the possible bias in the best-fit $R_{{\rm gas}_*}$ due to the imposed r₀ selection, and found that the best-fit $R_{\rm gas_*}$ was insensitive to the choice of r₀. Namely, they obtained a consistent estimate of $R_{\rm gas_*}$ for a smaller r₀. However, choosing a smaller r₀ reduced the already small sample further and the parameters became poorly constrained.

In the present analysis, we have included galaxy–QSO pairs with impact separation as large as ρ ≈ 120 h⁻¹ kpc (Figure 3). The best-fit gaseous radius of $R_{\rm gas_*}=75\ h^{-1}$ kpc is significantly below the maximum separation in the pair sample, although the majority of the data (galaxies with intrinsic colors consistent with present-day disks and irregular's/starbursts) lie at ρ < 75 h⁻¹ kpc. To examine whether the best-fit $R_{\rm gas_*}$ is sensitive to the range of impact parameter covered by the pair sample, we repeat the likelihood analysis adopting only pairs with ρ < 75 h⁻¹ kpc. We find that the best-fit results remain the same.

In summary, we find based on a sample of 71 “isolated” galaxies that the extent of cool gas as probed by the Mg ii absorption features scales with galaxy B-band luminosity according to R_gas ∝  L^0.35_B, consistent with earlier results obtained using smaller samples. As discussed in Chen & Tinker (2008), the scaling relation, when combined with a fiducial mass-to-light ratio determined from galaxy halo occupation studies of wide-field galaxy survey data, indicates that the gaseous radius is a constant fraction of the halo radius over the mass scale of M_h = 10^10.6–10¹³ h^-1 M_☉ sampled in our observations.⁸ In addition, we find that the residuals of observed and best-fit W_r(2796) do not correlate with galaxy absolute B-band magnitude, implying a declined cold gas fraction with increasing halos mass in the mass regime probed by our sample.

The presence of a finite extent R_gas of Mg ii absorbing gas is similar to what is found for extended C iv (Chen et al. 2001) and also for O vi absorbing gas recently published by Chen & Mulchaey (2009) around galaxies. The origin of such a finite boundary for metal-line absorbers can be interpreted as due to a halo fountain phenomenon (cf. Bregman 1980), according to which the gaseous radius is driven by the finite distance the outflowing material can travel from an early episode of starburst. We find this scenario unlikely because of the broad range of intrinsic colors covered by our galaxy sample, but a detailed stellar population synthesis to investigate the star formation history will shed more light on this issue.

Alternatively, the finite boundary of absorbing clouds probed by these metal-line absorbers can be understood as a critical radius below which cool clouds can form and survive in an otherwise hot medium. This two-phase model to interpret QSO absorption-line systems was formulated in Mo & Miralda-Escudé (1996) and later re-visited by Maller & Bullock (2004). This is also shown in recent high-resolution numerical simulations of Milky-Way type halos, in which accreted cold streams are disrupted by shocks but the remaining gas overdensities serve as the “seeds” necessary to form cool gaseous clouds through thermal instabilities within ≈1/3 of the halo radius (e.g., Kereš & Hernquist 2009).

The formation and presence of such cool clouds in a hot halo have several important implications for the studies of galaxy formation and evolution. For example, the confining hot medium around the cool clouds is expected to have a low density and long cooling time, reducing the “overcooling problem” in standard galaxy formation models (e.g., Maller & Bullock 2004; Kaufmann et al. 2009). In addition, these clouds may provide the fuels necessary to support continuous star formation near the center of galactic halos (e.g., Binney et al. 2000). Furthermore, the cool clouds together with the confining hot medium offer a reservoir for missing baryons in individual galactic halos (e.g., McGaugh et al. 2010). A nominal candidate for these cool clouds in our Milky-Way Halo is the high velocity clouds (HVCs) of neutral hydrogen column density $N({\rm H\,\mathsc {i}})>10^{18}$ cm^-2 seen in all-sky 21 cm observations. However, unknown distances of these HVCs prohibit an accurate measurement of their total gas mass (e.g., Putman 2006).

High-resolution spectra of strong Mg ii absorbers have revealed the multi-component nature of the absorbing gas, with W_r(2796) roughly proportional to the number of components in the system (e.g., Petitjean & Bergeron 1990; Prochter et al. 2006). These discrete Mg ii absorbing components are natural candidates of the condensed cool clouds within ∼75 h⁻¹ kpc radius of known galaxies. Interpreting the intrinsic scatter σ_c as due to Poisson noise in the number of cool clumps intercepted along a line of sight allows us to estimate the number of absorption clumps per galactic halo per sightline, n^clump. In this simple model, each absorber is characterized as W_r = n^clump × k, where k is the mean absorption equivalent width per absorbing component. Following Poisson counting statistics, the intrinsic scatter $\sigma _{c}\equiv \sigma _{c,W_r}/(W_r\,\ln \,10)$ is related to n^clump according to

$$\begin{equation} \sigma _{c} = \frac{1}{\ln \,10}\frac{\sqrt{n^{\rm clump}+1}}{n^{\rm clump}}. \end{equation} \tag{ 25 }$$

For the isothermal model, we have determined a best-fit intrinsic scatter of σ_c = 0.104, which leads to a mean number of absorbing clumps of

$$\begin{equation} n^{\rm clump} \sim 18 \end{equation} \tag{ 26 }$$

and a mean absorption equivalent width per clump of k = 0.04 Å at a median projected distance of 〈ρ〉 = 26 h⁻¹ kpc. The corresponding Mg ii absorbing column density per clump is $N({\rm Mg\,\mathsc {ii}})\approx 10^{12}$ cm^-2. We note that if other factors contribute to the observed σ_c, then the corresponding Poisson noise is smaller and the inferred n^clump will be bigger.

To estimate the total baryonic mass contained in these cool clouds traced by Mg ii absorbers, we first assume a mean clump size of 1 kpc (consistent with the size of H i clouds seen around M31; Westmeier et al. 2008), a mean metallicity of 1/10 solar (consistent with what is seen in the HVC Complex C; see Thom et al. 2008 for a list of references), and a mean ionization fraction of $f_{\rm Mg^+}=0.1$ for photo-ionized clouds (see Figure 6 of Chen & Tinker 2008). We derive a mean gas density of n_H ≈ 10⁻³ cm⁻³. Assuming a spherical shape for the clumps leads to a mean clump mass of M^clump_b = 1.7 × 10⁴ M_☉. Next, we adopt an isothermal density distribution of the clumps (Equation (16)) and derive a total number of clumps N^clump = 2 × 10⁵ h⁻² within R_gas = 75 h⁻¹ kpc. The total baryonic mass in these cool clumps is found to be M_b ∼ 3 × 10⁹ h⁻² M_☉, comparable to the total cold gas content seen in the Milky-Way disk (e.g., Flynn et al. 2006). The inferred total mass would be still larger if n^clump increases as the Poisson noise decreases.

If we further assume that it takes a free-fall time for the clumps to reach the center of the halo, we infer a cool gas accretion rate of $\dot{M}_{\rm gas}\sim 3\,{M}_\odot \,{\rm yr}^{-1}$. The inferred gas accretion rate is interestingly close to the star formation rate seen in the Milky-Way disk (e.g., Robitaille & Whitney 2010). However, we note two issues in this simple picture.

The first issue concerns the lifetime of these cool clumps relative to the time it takes for them to reach the inner few kpc of the host halo. Previous studies have shown that clouds of mass M_c ≲ 10⁵ M_☉ cannot survive thermal conduction and ram pressure as they move through a low-density, hot medium (see, e.g., Mo & Miralda-Escudé 1996; Maller & Bullock 2004; Heitsch & Putman 2009). To alleviate this problem, the clumps will need to have lower metallicities and/or bigger sizes than what we have assumed. For example, if we assume a mean clump size of 2 kpc and a mean metallicity of 0.05 solar, then we derive a mean clump mass of M^clump_b = 1.3 × 10⁵ M_☉. The total number of clumps would be N^clump = 5 × 10⁴ h⁻² and the total cool gas mass would be M_b ∼ 6.5 × 10⁹ h⁻² M_☉ at r ⩽ 75 h⁻¹ kpc.

Second, we note that the expected H i column density of these Mg ii absorbing clumps is $N({\rm H\,\mathsc {i}})\sim 10^{16}$ cm^-2, below the $N({\rm H\,\mathsc {i}})$ threshold provided by 21 cm observations. The H i content is therefore too low for these clumps to be the distant counterparts of the local HVCs. Nevertheless, the lack of strong (W_r(2796) ⩾ 0.5 Å) Mg ii absorbers beyond 30 h⁻¹ kpc (Figure 10) which have associated H i of $N({\rm H\,\mathsc {i}})> 10^{18}$ cm^-2 (Rao et al. 2006), constrains the distances of the Milky-Way HVCs at d ≲ 50 kpc. Within this close distance range, the implied total gas mass in the HVCs is expected to be M^HVC_b < 10⁹ M_☉ (Putman 2006).

The exercise presented above shows that the cool clouds that give rise to Mg ii absorbers and the implied presence of a hot confining medium can provide a substantial reservoir for missing baryons in individual galactic halos (see also Kaufmann et al. 2009). The presence of extended Mg ii absorbing halos around galaxies of a broad range of intrinsic colors argues against an outflow origin of these absorbers. Under an infall scenario, we show that these cool clouds provide sufficient fuels to support continuous star formation in their host galaxies.

The large covering fraction of Mg ii absorbing gas (Section 6.1) and the scaling relation between gaseous extent and galaxy B-band luminosity (Section 6.2) represent the empirical evidence supporting the halo occupation model developed in Tinker & Chen (2008, 2009; see also Gauthier et al. 2009). Here, we compare the results of our studies with recent surveys by other groups.

First, we have noted the discrepancy between the high gas covering fraction around ∼L_* galaxies and the low incidence of Mg ii absorbers around LRGs studied in Gauthier et al. (2010). We have interpreted the contrast as due to the different halo mass scales probed by the two galaxy samples. Cosmological simulations have shown that the growth of galaxies progresses primarily through stable accretion of intergalactic gas (e.g., Kereš et al. 2005; 2009). In low-mass halos (M_h < M^crit_h = 10^11.5 h^-1 M_☉), accretion proceeds to the center of the halo through dense cold streams (Dekel & Birnboim 2006; Kereš et al. 2005; 2009). As galaxies grow larger in mass (M_h > M^crit_h = 10^11.5 h^-1 M_☉), both numerical simulations and analytic models show that a progressively larger fraction of the accreted gas is shock heated to the virial temperature. Some fraction of the shock-heated gas may be able to cool in the inner region where gas density is high. The majority of the galaxies presented in this paper are on the transitional mass scale, M_h = 10^{11 − 12.5} h^-1 M_☉. In contrast, the LRGs studied by Gauthier et al. (2010) are in a higher-mass regime of M_h > 10¹³ h^-1 M_☉, where a diminishing amount of cold gas is expected because the cooling time is long. The differential gas covering fraction found in our galaxies and in those LRGs is therefore consistent with theoretical expectations.

Kacprzak et al. (2008) have also studied the extent and covering fraction of Mg ii absorbing gas based on a sample of 37 galaxies at z = 0.3–1 that are found in the vicinity of a known Mg ii absorber. While they did not see a clear correlation between the extent of Mg ii absorbing gas and galaxy luminosity, they found that for a scaling power of β = 0.18–0.58 and a characteristic gaseous extent of R_* = 56–105 h⁻¹ kpc the covering fraction of Mg ii absorbers with W_r(2796) ⩾ 0.3 Å is no more than 70%. Our best-fit isothermal model indicates that the characteristic gaseous radius at W_r(2796) = 0.3 Å is $R_{\rm gas_*}(W_r=0.3\,AA)=47\ h^{-1}$ kpc (Figure 11), less than the gaseous radius considered by Kacprzak et al. Excluding the outliers, we find a mean covering fraction of κ_0.3[ρ′ < 47 h⁻¹ kpc] ≈ 76% for Mg ii absorbers with W_r(2796) ⩾ 0.3 Å. Including the outliers, we find a mean covering fraction of κ_0.3[ρ′ < 47 h⁻¹ kpc] ≈ 78% for Mg ii absorbers with W_r(2796) ⩾ 0.3 Å. The gas covering fraction found in our sample is somewhat higher than the parameter space explored by Kacprzak et al., but this can be explained by the larger gaseous radius considered by these authors.

In addition, Barton & Cooke (2009) have recently carried out a search of coincident Mg ii absorbers in the vicinity of 20 luminous galaxies (M_r − 5 log  h ⩽ −20.5) identified at z ∼ 0.1 in the SDSS galaxy spectroscopic sample. The authors reported a low gas covering fraction of κ_0.3[ρ < 75 h⁻¹ kpc] ≲ 0.4 and κ_0.3[ρ < 35 h⁻¹ kpc] ∼ 0.25 for Mg ii absorbers of W_r(2796) ⩾ 0.3 Å. These results are inconsistent with the high gas covering fraction found in our survey. The design of the Barton & Cooke survey is very similar to ours presented here, with only small differences in the targeted redshift range (z ∼ 0.1) and luminosity scale (>L_*). The discrepancy in the observed gas covering fraction therefore requires further perusal.

A fundamental difference in our respective measurements is that Barton & Cooke excluded observed strong Mg ii absorbers in the covering fraction calculation if the QSO spectrum did not have sufficient S/N for identifying absorbers of W_r(2796) = 0.3 Å. Of the six galaxies at ρ ⩽ 35 h⁻¹ kpc, three have associated Mg ii absorbers of W_r(2796) = 1.93 ± 0.18 Å, W_r(2796) = 1.91 ± 0.10 Å, and W_r(2796) = 0.71 ± 0.15 Å, respectively, and three do not have corresponding Mg ii absorbers to 3σ limits of W_r(2796) < 0.3 Å. The authors excluded from the covering fraction calculation two strong absorbers, W_r(2796) = 1.93 ± 0.18 Å and W_r(2796) = 0.71 ± 0.15 Å, and derived κ_0.3[ρ < 35 h⁻¹ kpc] ∼ 0.25 based on the remaining four galaxies.

We note that the goal of our study is to determine the incidence of Mg ii absorbers at the location of a known galaxy. Absorbers that are observed to have W_r(2796) = 0.71 ± 0.15 Å or W_r(2796) = 1.93 ± 0.18 Å confirm the presence of such absorbing gas in large quantities, despite the fact that the QSO spectrum may not offer the sensitivities required for uncovering weaker absorbers of W_r(2796) ∼ 0.3 Å at high confidence levels. Excluding these sightlines imposes a bias that would result in an underestimate of the gas covering fraction.

We have retrieved from the SDSS data archive the galaxy photometric and spectroscopic data in Barton & Cooke (2009), and determined the rest-frame B-band absolute magnitude and B_AB − R_AB color for each of the 20 galaxies according to the procedures described in Section 4.1. The rest-frame absolute B-band magnitudes of these galaxies span a range from M_B − 5log  h = −19.6 to M_B − 5log  h = −20.7 with a median of 〈M_B − 5log  h〉 = −20.3. The rest-frame B_AB − R_AB colors range from B_AB − R_AB ≈ 0.6 to B_AB − R_AB ≈ 1.2 with a median of 〈B_AB − R_AB〉_med ≈ 1.0. Figure 13 displays the observed W_r(2796) versus ρ relation for the Barton & Cooke sample, superimposed on top of our own data. We have converted the 3σ upper limits published by these authors to 2σ upper limits in order to be consistent with our own measurements.

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Figure 13 shows that with the exception of two outliers at ρ′ ≈ 25 h⁻¹ kpc, the Barton & Cooke sample is in general agreement with what is seen in our sample. Six of the 12 upper limits in the Barton & Cooke sample do not have sufficient sensitivities for constraining the presence/absence of Mg ii absorbing gas at the level of W_r(2796) ∼ 0.3 Å. In addition, the galaxy at $\log \,\rho ^{\prime }=\log \,\rho +0.14\,(M_B-M_{B_*})\approx 1.6$ with a strong Mg ii absorber of W_r(2796) = 1.24 (SDSSJ144033.82+044830.9) was later found to have a companion galaxy at ρ = 18.6 h⁻¹ kpc. The absorber would therefore be considered as in a “group” environment. The galaxy at ρ′ ≈ 20 h⁻¹ kpc with a 2σ upper limit of W_r(2796) < 0.13 Å (SDSSJ151541.23+334739.4) has a luminous companion (SDSSJ151536.18+334743.6) at projected distance <100 h⁻¹ kpc and velocity separation Δ v ≈ 27 km s⁻¹ away. This pair also represents a “group” that we have treated separately in our analysis.

Considering only pairs between “isolated” galaxies and QSOs for which sensitive limits for the corresponding Mg ii absorption are available leads to a sample of 12 galaxies with sensitive constraints for the corresponding Mg ii absorption features. We find three of the four galaxies at ρ′ < 35 h⁻¹ kpc have associated Mg ii absorbers of W_r(2796) > 0.3 Å, yielding κ_0.3[ρ′ < 35 h⁻¹ kpc] ≈ 0.75. We find seven of the 12 galaxies at ρ′ < 75 h⁻¹ kpc have associated Mg ii absorbers of W_r(2796) > 0.3 Å, yielding κ_0.3[ρ′ < 75 h⁻¹ kpc] ≈ 0.58. These values are consistent with our estimated gas covering fraction to within 1σ uncertainties. We therefore conclude that the observations presented in Barton & Cooke (2009) are consistent with the high gas covering fraction seen in our sample.

Finally, Zibetti et al. (2007) studied the nature of Mg ii absorbing galaxies at z = 0.37–1 based on stacked images of QSOs with known foreground Mg ii absorbers of W_r(2796) > 0.8 Å. After subtracting the QSO point spread function in the stacked images, these authors obtained an angular averaged image of Mg ii absorbing galaxies out to 100 kpc. They found that the luminosity-weighted mean colors of the extended emission are consistent with present-day intermediate-type galaxies. In addition, the authors found that weak absorbers of W_r(2796) < 1.1 Å originate primarily from red passive galaxies while stronger absorbers display bluer colors consistent with star-forming galaxies. Based on the differential color distribution, Zibetti et al. argued that the origin of strong Mg ii absorbers might be better explained by models of metal-enriched gas outflows from star-forming/bursting galaxies.

Our analysis in Section 5.3 shows that the observed Mg ii absorber strength does not depend strongly on galaxy intrinsic color, which appears to contradict the observations of Zibetti et al. To understand the discrepant results, we first recall based on the observed W_r(2796) versus ρ anti-correlation in Figure 8 that the mean impact parameter of the absorbing galaxies for weak absorbers is expected to be larger than that for stronger absorbers. In addition, Figure 8 also shows that for a sample of randomly selected QSO–galaxy pairs in our study, the fraction of red galaxies is found to be a factor of 2 larger at larger impact parameters ρ>30 h⁻¹ kpc than at closer distances. This is not due to selection biases, because we did not preferentially select blue galaxies at smaller impact parameters. Instead, this can be understood by considering the fact that redder galaxies are on average more luminous and have a lower space density than bluer and on average fainter galaxies (e.g., Faber et al. 2007). The increased incidence of red galaxies at larger impact parameter, combined with the luminosity-weighted nature of the image stacking procedure, is expected to skew the stacked images of weaker absorbers toward redder colors. Whether or not this is sufficient to explain the observed color gradient between strong and weak Mg ii absorbers in the Zibetti et al. sample requires a detailed simulation that takes into account such intrinsic weighting by the space density and luminosity of the galaxies. Because such analysis has not been performed, it is not clear whether the respective results are inconsistent with each other. Nevertheless, the conclusion in favor of strong Mg ii absorbers being better explained by starburst outflows based on the observed color gradient should be viewed with caution.