SCATTERED EMISSION FROM z ∼ 1 GALACTIC OUTFLOWS

Table 1 lists the measured properties of 32016857. It has notably blue U − B color. The SFR estimated from the [O ii] luminosity, as described in the notes to Table 1, is approximately 80 M_☉ yr⁻¹. The mass derived from the spectral energy distribution fit, log (M/M_☉) = 9.82 (Bundy et al. 2006), places 32016857 in the lowest tertile of the LRIS sample by stellar mass. This mass estimate should be accurate to within a factor of two because the K-band photometry of 32016857 directly constrains the rest-frame, near-IR continuum. At z ≈ 1, abundance matching places galaxies with stellar mass log M_*/M_☉ = 9.8 in halos of total mass log M_h/M_☉ ≈ 11.5 (Behroozi et al. 2010), consistent with the clustering of DEEP2 galaxies which places galaxies brighter than M_B < −20.77 in halos more massive than log M_h/M_☉ = 11.3 (Coil et al. 2008). In numerical simulations, halos with masses similar to our best estimate for 32016857 (specifically M_h < 10¹² M_☉), gas accretion is dominated by the cold flows never shock-heated to the halo virial temperature (e.g., Kereš et al. 2005).

In the red side spectrum, shown in Figure 1(b), the resolution of 220 km s⁻¹ FWHM does not quite resolve the prominent [O ii] λλ3727.09, 3729.88 emission lines. The blue spectrum, 282 km s⁻¹ FWHM resolution, reveals strong emission in the Mg ii λλ2796.35, 2803.53 doublet with a P-Cygni profile (bottom panels). The recombination lines from hydrogen and helium as well as the strong emission in the C ii] λ2324 − 29, C iii] λ1908.73, [Ne iii] λ3869.85, λ3968.58, and [O ii] lines come from H ii regions photoionized by massive stars. The absence of [Ne v] λ3426.98 emission argues against an active galactic nucleus (AGN) as the dominant source of ionizing radiation.

The emission lines in the red spectrum dominate the cross-correlation signal with template spectra and thereby determine the LRIS redshift of z = 0.9392. The root-mean-square (rms) error in the dispersion solution of the red spectrum is only a few  km s⁻¹; hence the rms error in the blue dispersion solution, about 18 km s⁻¹, determines the systematic uncertainty in the relative velocities of the low-ionization absorption and nebular emission. In the blue spectrum, the centroid of the C iii] emission line lies just −30 ± 31 km s⁻¹ from the intercombination transition which dominates the emission at high density. We readily acknowledge a larger discrepancy of 50 km s⁻¹ between the LRIS redshift and the DEEP2 survey redshift, likely caused by differences in slit position angle but irrelevant to the value of relative velocities between the absorption lines and nebular emission.

The absorption lines in the LRIS spectrum are not sufficiently resolved to measure the maximum blueshift from the Fe ii line profiles. The fitted centroid of the Fe ii absorption lines is blueshifted V₁ = −91 ± 15 km s⁻¹ (Martin et al. 2012). The projected outflow velocity is more accurately estimated by the two-component fit shown in the right column of Figure 2. Correcting for the interstellar absorption at the systemic velocity increases the blueshift of the Doppler component to V_Dop = −227 ± 82 km s⁻¹.

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While the λ2803 absorption trough may be filled in by emission from both Mg ii transitions, only λ2796 emission fills in the intrinsic λ2796 absorption trough. In Mg ii λ2803, Figure 1 shows absorption blueward of the systemic velocity and redshifted emission. The marginally resolved emission wing extends to +310 km s⁻¹ (Martin et al. 2012). The corresponding red wing of the Mg ii λ2796 profile may significantly fill in the bluest portion of the intrinsic λ2803 absorption profile. Emission at the systemic velocity in both transitions likely fills their intrinsic absorption profiles at the systemic velocity.

Optically thick Mg ii absorption will produce an intrinsic λ2803 trough with the same equivalent width as the λ2796 trough which is twice as strong in the optically thin limit. Remarkably, however, the absorption equivalent width of the Mg ii λ2796 absorption trough is less than that in λ2803 as seen in both Figure 1 and Table 2. While low signal-to-noise ratio (S/N) could produce such an effect through measurement error, our analysis suggests a different interpretation. We can understand the net Mg ii profile provided the intrinsic ratio of emission-line strengths W_em(λ2796)/W_em(λ2803) exceed the ratio of absorption-line strengths W_abs(λ2796)/W_abs(λ2803). In support of this approach, we note that in Table 2 the net emission equivalent width of the stronger Mg ii transition, −2.13 ± 0.29 Å, is higher than in the weaker Mg ii line at λ2803. Qualitatively at least, we can therefore create a line profile with a weak λ2796 trough (relative to λ2803) if the intrinsic absorption troughs are both saturated and therefore equal in area.

To explore this solution quantitatively in a manner that fully incorporates the spectral S/N, we fit a model consisting of an emission doublet simply added to the intrinsic absorption troughs. This model assumes that the emission region lies beyond most of the absorbing gas. The caption of Figure 3 describes the functional form of these components. We wrote a custom Markov Chain Monte Carlo fitting code in Python to explore the range of possible parameter values. The fit shown in Figure 3 illustrates a typical fit drawn from the Markov chain.

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To estimate the values of the absorption and emission components, we used the posterior probability distributions compiled for each parameter from the full Markov chain to determine the mean values of the parameters and their 1σ uncertainties. Our fit describes the absorption troughs with a Doppler shift of $V_{{\rm abs}} = -11^{+51}_{-49}$ km s⁻¹, Doppler parameter $b_{{\rm Dop}} = 81^{+23}_{-24}$ km s⁻¹, covering factor $C_f = 0.91^{+0.06}_{-0.12}$, and optical depth at line center $\tau _0 = 68^{+80}_{-44}$; it finds an emission component with a redshift of $V_{{\rm em}} = 105^{+12}_{-10}$ km s⁻¹, width $\sigma = 14^{+14}_{-9}$ km s⁻¹, amplitude $A_{2796} = 4.74^{+0.77}_{-0.81}$ for the λ2796 transition, and doublet ratio $A_{2796} / A_{2803} = 1.51^{+0.20}_{-0.16}$. We find that the intrinsic absorption troughs typically have very similar equivalent widths. In contrast, only doublet ratios A₂₇₉₆/A₂₈₀₃, near 1.5 provide acceptable descriptions of the emission. We conclude that the optical depth of the emission region in our slit is lower than that of the absorbing gas along our sightline. We interpret the difference between the absorption and emission doublet ratios as an indication that different physical regions of the galaxy produce the emission and absorption—e.g., non-spherical geometries for the scattering region and/or significant Mg ii emission from H ii regions.

The net Fe ii absorption troughs more nearly reflect the intrinsic shape of the absorption profile than does Mg ii absorption. The resonance emission that fills in these troughs may be generated by H ii regions or created by the absorption of continuum photons in the surrounding gas. As emphasized in Erb et al. (2012), H ii regions produce much more emission in Mg ii than in Fe ii, so nebular emission will have a proportionately larger impact on the shape of the net Mg ii line profile. In our integrated spectra of 32016857, the flux ratio of the [O ii] and Mg ii emission, F([O ii]$\lambda \lambda 3726,29) / F({\rm Mg\scriptsize{II}} \lambda 2796) \approx 42$ (uncorrected for reddening), is consistent with line ratios calculated for nebulae at slightly sub-solar metallicity photoionized by a starburst spectral energy distribution. For example, comparison of this line ratio to Figure 16 in Erb et al. (2012) indicates an ionization parameter log U ≳ −2.9, where the lower limit indicates that the reddening-corrected line ratio would require a larger ionization parameter. In addition, in contrast to the ground state of Mg ii, which is a singlet, the ground-state of Fe ii has fine structure. In Fe ii, absorption of a resonance photon is often followed by the emission of a longer wavelength photon (Rubin et al. 2011; Prochaska et al. 2011; Martin et al. 2012). This fluorescent emission, clearly visible in Figure 2 near the systemic velocity, does not fill in the resonance absorption troughs.

Table 2 summarizes the equivalent widths and velocities of Fe ii absorption troughs and Fe ii* emission. The relative strengths of the individual absorption lines are not well fit with standard curve-of-growth techniques due to differential amounts of emission filling. Using the two-component fit shown in Figure 2, Martin et al. (2012) estimate the ionic column density of the Doppler component lies in the range log [N_Dop(Fe⁺)C_f(cm⁻²)] = 14.42–15.53, where C_f is a gas covering factor of the order of unity.

The 2D spectra of 32016857 constrain the spatial extent of the emission regions. Figure 4 compares the structure of the Mg ii and [O ii] line emission. While the Doppler shift of the [O ii] emission shifts by approximately 190 km s⁻¹ over 17 kpc along the slit, no velocity gradient is detected in Mg ii emission. While the H ii regions producing the [O ii] emission probably contribute to the total Mg ii emission, based on the kinematic differences between the 2D spectra, we suggest that the Mg ii emission takes on the velocity of the outflowing material, not the velocity of the H ii regions, because it is scattered by halo gas.

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Figure 5 compares the emission profiles along the slit. The continuum emission, rather than showing the symmetric profile of an inclined disk, shows an extended feature to the west with strong [O ii] but no Mg ii emission. The [O ii] emission is unresolved to the east. In contrast to this nebular line emission, the Mg ii surface brightness profile is spatially extended to the east but unresolved to the west. The opposite directions of the extended [O ii] and Mg ii emission along the slit suggest distinct physical origins.

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The Mg ii surface brightness averaged over a spatial resolution element (6 pixels) 10.6 kpc eastward of the galaxy is 3.1 standard deviations above the background. The continuum emission is also spatially extended to the east. Averaged over the same spatial resolution element at 10.6 kpc, the line emission is marginally, 2.1σ, more significant than the continuum emission indicating the line emission is likely more extended than the continuum emission. The Mg ii emission is not spatially resolved to the west.

To estimate the radial extent of the Mg ii emission, we introduce a simple model for the surface brightness profile measured along the slit inspired by, but different in detail from, the approach used by Rubin et al. (2011) to model the galaxy TKRS 4389. The emission model includes an H ii region component and an extended component. The normalized [O ii] surface brightness profile determines the shape of the H ii component. Since the observed Mg ii profile is spatially resolved to the east but not to the west, we require an asymmetric extended component. We adopt a semi-circular emission region of constant surface brightness. The radius, R_C, of the extended emission describes the projected distance from the center of 32016857, and the straight-side of the semi-circle is perpendicular to the slit. This 2D model of the surface brightness profile is convolved with a Gaussian model of the atmospheric seeing, FWHM = 08 and then "observed" through a 12 slit. In analogy with Lyα radiative transfer, photons scattered off the far side of the outflow escape as redshifted emission.

We explore 14 values of R_C from 4.9 to 35.9 kpc, corresponding to an angular extent of 0621 to 4554 along the slit. For each model of the extended emission, we fit the Mg ii surface brightness profile with a linear combination of the H ii and extended components. Figure 6 shows the minimum residuals at R = 11.4 kpc, where the separation between models is about 0.34 kpc. The extended component and the H ii component determine the net surface brightness profile to the east and west, respectively, in Figure 6. For the best-fit model, the eigenvalues of the two components are similar, and the extended component contributes 46% of the total Mg ii flux. While the residuals show that this model provides a reasonable description of the data, it does not tightly constrain the radius of the scattering halos. Smaller halos cannot be excluded with high confidence due to the profile smearing caused by atmospheric turbulence, and this blurring also means the reduced chi-square values increase slowly with increasing halo radius.

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The bluest fifth of the Martin et al. (2012) sample contains 42 galaxies with U − B < 0.459 and is of particular interest since the Mg ii emitters tend to have blue colors (Martin et al. 2012; Kornei et al. 2013). All but 4 of these spectra cover the Mg ii doublet, yet only 10 of these 38 show prominent Mg ii emission in integrated spectra. Among this subsample, we found only two examples of spatially extended emission in the Mg ii doublet. In addition to 32016857, the Mg ii emission in our spectrum of 32010773 is spatially extended.

Figure 7 shows the surface brightness profiles of the Mg ii emission (panel (d)), blue continuum, [O ii] emission (panel (c)), and red continuum along the slit crossing 32010773. In contrast to the surface brightness profiles measured across 32016857, the Mg ii emission across 32010773 is no more extended than the blue continuum emission. In the continuum-subtracted, 2D spectra, the velocity gradients of the Mg ii and [O ii] emission along the slit do not obviously differ, so the Mg ii emission may come directly from H ii regions. Our data do not require (nor do they rule out) scattered Mg ii emission. This galaxy does clearly have an outflow; the large blueshift of the Fe ii resonance lines, V₁ = −198 ± 46 km s⁻¹, is quite significant. The color of 32010773, U − B = 0.23 is among the bluest in the sample; but the luminosity of 32010773, M_B − 5log h₇₀ = −20.68, is much lower than the luminosity of either 32016857 or TKRS 4389. The absence of resonance emission that is clearly extended relative to the continuum around 32010773 is consistent with the conjecture that detection of a scattering halo requires especially luminous galaxies.

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In addition to these 10 Mg ii emitters with very blue color, Kornei et al. (2013) identified 12 more Mg ii emitters in the Martin et al. (2012) sample with U − B redder than 0.459. Among these redder Mg ii emitters, we found spatially extended Mg ii and continuum emission across 12019973. Figure 8 shows the variation in the Mg ii and [O ii] emission along the slit. Across 12019973, much like 32010773, the surface brightness profile of the line emission is no more extended than the continuum emission, neither proving nor ruling out scattering by an outflow. We note that the Doppler shift of the Fe ii resonance lines, V₁ = −59 ± 28 km s⁻¹, provides marginal evidence for an outflow from this galaxy. The color, U − B = 0.539, and stellar mass, log (M/M_☉) = 10.15, of 12019973 are typical of the sample. This galaxy has a relatively low luminosity, M_B − 5log h₇₀ = −20.57, which may contribute to the faintness of any extended halo.

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Figure 9 shows the 2D spectra of another Mg ii emitter, 22028686, a star-forming galaxy with U − B > 0.459. Although the Mg ii emission is not spatially resolved, the resonance absorption troughs extend well beyond the emission region. The Mg ii absorption lines are tilted in the opposite sense of the velocity gradient indicated by the [O ii] Doppler shifts along the slit. Since the [O ii] emission defines the systemic redshift, the redshifted and blueshifted [O ii] emission to the northeast and southwest of the galactic center reveal either the projected rotation of the galaxy or the orbital motion of two merging galaxies. The centroid of the Mg ii absorption to the southwest of the galaxy is consistent with the velocity of the blueshifted side of the galaxy. It is the even larger blueshift of the Mg ii absorption centroid to the northeast that makes the absorption lines tilt in the opposite direction of the [O ii] emission. Where the Mg ii absorption is blueshifted on the northeastern side of the galaxy, we detect strong resonance emission at redshifted velocities. No resonance emission is detected along the southwestern half of the slit. We note that the Fe ii absorption in the integrated spectrum does not show a net Doppler shift, V₁ = −1.9 ± 9.2 km s⁻¹. Much of the shift in the centroid of the Mg ii absorption may therefore be caused by variations in emission filling along the slit, and the spatial gradient in the absorption velocity likely does not reflect galactic rotation.

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The detection of Mg ii emission from just a portion of the absorbing region in 22028686 may help illuminate the physical conditions required for the resonance emission to escape the galaxy. To gain this insight, we review studies which have mapped Na i resonance absorption across galaxies. As illustrated, for example, by spectral mapping of NGC 1808 (Phillips 1993) local ULIRGs (Martin 2006), and Mrk 231 (Rupke et al. 2005; Rupke & Veilleux 2011), the absorption from outflows is not redshifted anywhere across the galaxy because the absorbing gas must lie in front of the continuum source. Across local ULIRGs, the Na i velocity gradient is very similar to the slope of the rotation curve, but the absorption trough is highly blueshifted everywhere along the slit. This situation would be expected if the outflow was launched from a rotating disk, and we might expect a small velocity gradient in the absorption Doppler shift (not seen) comparable to the tilt of the nebular emission lines. In spite of the very limited spatial resolution across 22028686, we speculate that the geometry might instead mimic NGC 1808 where differences in extinction shape the resonance line profile of the outflow. As illustrated in Figures 1 and 4 of Phillips (1993), the blueshifted lobe of the outflow is detected in absorption across the inclined disk. Resonance emission is detected from the far side of the outflow where it pokes out from behind the disk but plumes of dust visible on the near side of the outflow at the location of the slit presumably destroy the resonance photons. Hence the implication from the 22028686 spectrum seems to be that the sightline toward the absorbing gas on the northeast side encounters less dust than that toward the southwest side of the slit. Higher resolution imaging could confirm the presence of a disk, determine the disk orientation, and perhaps reveal the locations of dust filaments in this galaxy.

In summary, among 145 spectra with coverage of the Mg ii doublet, including 50 galaxies with either prominent Mg ii emission or very blue U − B color, we find only three examples of spatially resolved Mg ii emission: 32016857, 32010773, and 12019973. Our analysis of 32016857 doubles the number of known galaxies with Mg ii emission extending beyond the stellar continuum. With only two examples where the extended emission must be scattered, we can only begin to speculate about the conditions producing scattered Mg ii halo emission. However, since the unique properties of 32016857 and TKRS 4389 are their high luminosities (for their blue color), we suggest that large luminosities are essential to raise the surface brightness of the scattered emission to detectable levels. Further support for the idea that most scattering halos are below current detection limits comes from stacking the surface brightness profiles of 1 < z < 2 Mg ii—emitting galaxies; their Mg ii emission is marginally more extended than the near-UV continuum at radii of 08 (Erb et al. 2012).

Our spectrum of 32016857 shows prominent fluorescent emission in Figures 1 and 2. We did not find extended emission in resonance lines of Fe ii or fluorescent Fe ii* lines and neither did Rubin et al. (2011) in their TKRS 4389 spectrum. The Fe ii λ2383 absorption is optically thick in most galaxy spectra; under spin-orbit (L-S) coupling, the only permitted decay from the upper energy level, here the excited $z^6F^0_{11/2}$ state of Fe⁺, is to the ground a⁶D_9/2 state. The absence of Fe ii λ2383 emission in spectra with prominent Mg ii λ2803 emission is at first surprising considering the similar oscillator strengths of the λ2383 and λ2796 transitions and the (only) 20% larger cosmic abundance of magnesium (relative to iron).

Several factors may contribute to the paucity of Fe ii λ2383 emission. A greater depletion of iron (relative to magnesium) onto grains, for example, may reduce the Fe ii λ2383 optical depth relative Mg ii λ2803. In the Milky Way, however, the relative depletion of Fe and Mg in the halo clouds is much lower than the ratio measured for disk clouds (Savage & Sembach 1996). Hence, we might anticipate a similar depletion of Fe and Mg onto grains in outflows. Alternatively, since iron is not a light element, deviations from pure L-S coupling might in principle allow decay to another term thereby reducing the resonance emission (J. M. Shull 2012, private communication), but we find no observational evidence for unidentified emission lines in support of this conjecture in the near-UV spectra. Third, as suggested recently by Erb et al. (2012), boosting the Mg ii emission, rather than suppressing the Fe ii emission, offers another explanation. Those authors show that photoionized gas produces more emission in Mg ii than Fe ii, so emission from H ii regions may boost the intensity of the scattered Mg ii radiation relative to the scattered Fe ii photons.

The most important factor driving this discrepancy, however, appears to be the ionization corrections. Giavalisco et al. (2011) show (see their Figure 14) that photoionization models of very low column density (N_H ∼ 10¹⁸ cm⁻²) gas indicate a large plausible range in the relative column density; $N_{{\rm Mg\,\scriptsize{II}}} / N_{{\rm Fe\,\scriptsize{II}}}$ increases from approximately unity (log U = −4) to just over 100 at log U = −2. Even though the first (7.646 eV, 7.870 eV) and second (15.035 eV, 16.18 eV) ionization potentials of Mg and Fe, respectively, are quite similar, the Fe is mainly in Fe⁺³ while the Mg is in Mg⁺⁺ in our photoionization models, which we describe in the next section. This happens because it takes only 30.6 eV to ionize Fe⁺⁺ to Fe⁺³ (and 54.8 eV to make Fe⁺⁴) but 80 eV to remove another electron from Mg⁺⁺.

We model the outflows with a radial velocity field. For 32016857, we picture a scenario where our slit intersects the outflow cone on only the west side of galaxy. In TKRS 4389, in contrast, the long slit observation resolves Mg ii emission on both sides of the slit (Rubin et al. 2011). A simple model with a constant surface brightness halo suggests the impact parameter of the scattering surface is 8.25 < b(kpc) < 13 for TKRS 4389 (Rubin et al. 2011) and roughly 11.4 kpc from 32016857.

The interaction region where a resonance transition such as Mg ii λ2796 can be scattered depends on the line profile. To simplify the calculation, we consider a delta function line profile centered on the resonance frequency ν₀ (in the reference frame of the bulk flow). Photons emitted by the starburst at frequency ν propagate radially until they encounter a gas parcel with radial velocity v(r_S) = c(ν − ν₀)/ν₀. We call the interaction spot the Sobolev point, r_S. Provided velocity v, μ, and the radial velocity gradient change slowly around r_S, the Sobolev expression (Sobolev & Gaposchkin 1960; Lamers & Cassinelli 1999) describes the scattering optical depth,

$$\begin{eqnarray*} &&\tau ^S_{\nu _0} = \frac{\pi e^2}{m_e c} f \lambda _0 n_i \left((1 - \mu ^2) \frac{v(r)}{r} + \mu ^2 \frac{d v}{d r} \right)^{-1}, \end{eqnarray*}$$

where f represents the transition oscillator strength.⁷ The incident photons and the velocity of the bulk flow are both radial, so the cosine of the angle θ between them takes the value μ = cos θ = 1. The frequency of the photons absorbed from the starburst varies only with v(r) for a spherical outflow.

With μ = 1 in Equation (3), the Sobolev optical depth for the stronger Mg ii transition is given by

$$\begin{eqnarray} &&\tau _{2796}^S (r) = 4.57 \times 10^{-7} \,{\rm cm}^{3}\,{\rm s}^{-1} n_{{\rm Mg}^+}(r) \left|\frac{dv }{dr} \right|_{r_S}^{-1}. \end{eqnarray} \tag{ 3 }$$

At some radius along the sightline at impact parameter b ≈ 11.4 kpc, we expect the scattering optical depth for the starburst photons to be approximately unity. We estimate the gas density by setting $\tau _{2796}^S(r) \approx 1$ and obtain

$$\begin{eqnarray} &&n_{{\rm Mg}^+} \approx 1.4 \times 10^{-9}\,{\rm cm}^{-3} \left|\frac{dv}{230\,{\rm km\,s}^{-1}} \frac{11.4\,{\rm kpc}}{dr} \right|_{r_S}. \end{eqnarray} \tag{ 4 }$$

If the sensitivity of our spectrum, rather than the radius of the last scattering surface, determines the observed extent of the scattered emission along the slit, then the optical depth (and the inferred density) would be higher than this estimate. However, we think modeling the spatial extent with the last scattering surface is roughly correct because the emission is not optically thick at b ≈ 11.4 kpc; at the largest impact parameters where Mg ii emission is detected, only λ2796 is detected, consistent with a two-to-one flux ratio and an optical depth in λ2803 less than unity. The gas density estimated from Equation (4) depends on the velocity gradient at the Sobolev point, and this radius is not uniquely determined from the impact parameter b alone.

Figure 10 illustrates the geometry for the outgoing, scattered photons at impact parameter b. Along a sightline, the angle θ ranges from 90° where the flow is perpendicular to the sightline to a minimum, θ_min, and maximum θ_max ≡ 180° − θ_min, constrained by the Doppler shifts of the absorption and emission lines. The Doppler shift of a scattered photon reflects the projected velocity component, v_z, and will be most blueshifted (redshifted) as θ_min → 0° (θ_max → 180°). These extrema mark the scattering that occurs at the largest radius along a sightline; and we infer the largest (smallest) radius from the lower (upper) limits on θ_min. The error bars on the angle come from considering the projection of the radial velocity (measured in absorption) onto to the line-of-sight component measured at impact parameter b in emission. For 32016857, we estimate 38° ⩽ θ_min ⩽ 75°, where the spectral fitting constrains the most blueshifted absorption ((|V_Dop| + b)) to the range 185–383 km s⁻¹; and the redshifted emission (V_z + σ) extends to at least 100–145 km s⁻¹. The maximum radius from which we detect scattered emission is therefore between 1.04b ≈ 12 kpc and 1.61b ≈ 18 kpc.

We estimate an average velocity gradient of 230 km s⁻¹ over 12 to 18 kpc. Substituting these values into Equation (4), we infer an ionic density between 8.9 × 10⁻¹⁰ cm⁻³ for r_S = 18 kpc (θ_min = 38°) up to 1.3 × 10⁻⁹ cm⁻³ for r_S = 12 kpc (θ_min = 75°). A specific form for the acceleration of the outflow, $v(r) = v_0 \sqrt{\ln r/R_0}$, demonstrates how a more physical model might change our density estimate. The acceleration of a momentum-driven outflow begins rapidly and then slows down with increasing distance. This model reaches a characteristic velocity v₀ at a radius a few times the launch radius, r = 2.72R₀. Choosing R₀ such that v(r_S) = v₀ and r_S = 2.72R₀, we obtain dv/dr = 0.5v₀/r_S. Placing the Sobolev point at 2.72R₀ for comparative purposes, the estimated velocity gradient at r_S takes a value half as large as the constant acceleration limit; and our density estimate would decrease by a factor of two.

For TKRS 4389, Rubin et al. (2011) measured the Doppler shift of the low-ionization absorption lines and fit a radius for the scattering halo to the surface brightness profile of the Mg ii emission. Substituting their values of V_Dop ≈ 200–300 km s⁻¹ and b ≈ 8.25–12.4 kpc into Equation (4), we estimate n_{Mg +} ≈ 1.1–2.6 × 10⁻⁹ cm⁻³. Since the Sobolev radius could lie at radii as large as r_S ≈ 1.5b, where we have used the Doppler shift of the λ2803 emission, v_z ≈ 226 km s⁻¹ in Table 1 of Rubin et al. (2011), and v = 300 km s⁻¹ to estimate θ_min = cos ⁻¹(v_z(r_S)/v(r_S)) ≈ 41°, we estimate a lower limit on the ionic density of roughly 0.73 × 10⁻⁹ cm⁻³. The slightly larger density, n_{Mg +} ≈ 7 × 10⁻⁹ cm⁻³, found by Rubin et al. (2011) is consistent with substituting the −800 km s⁻¹ Doppler shift of the blue wing on the Mg ii profile into Equation (4) along with the minimum Sobolev radius. These arguments constrain the ionic density at the Sobolev radius in the TKRS 4389 outflow to the range 0.7–7 × 10⁻⁹ cm⁻³.

Given the ionic density $n_{{\rm Mg}^+}$, the total gas density scales inversely with the ionization fraction, metallicity, and dust depletion. Defining the ionization fraction

$$\begin{eqnarray} &&\chi ({\rm Mg^+}) \equiv n_{{\rm Mg^+}} / n_{{\rm Mg}}, \end{eqnarray} \tag{ 5 }$$

the hydrogen number density is

$$\begin{eqnarray} &&n_{{\rm H}} = \frac{n_{{\rm Mg^+}} }{\eta ({\rm Mg}) d({\rm Mg}) \chi ({\rm Mg^+}) }. \end{eqnarray} \tag{ 6 }$$

We scale to solar metallicity, η(Mg) = 3.8 × 10⁻⁵, which is appropriate (to a factor of two or better) for a wind loaded with interstellar gas from a galaxy with the stellar mass of the 32016857 (Zahid et al. 2011). We chose a fiducial value for the depletion of Mg onto grains typical of clouds in the Milky Way disk, d(Mg) ≈ 6.3 × 10⁻² (Savage & Sembach 1996), and note that a lower depletion similar to clouds in the Milky Way halo (d ≈ 0.28) would lower our density estimate by a factor of four.

To estimate χ(Mg⁺), we start with a rough assumption, which we relax later, namely that the bulk of the Mg ions are either Mg⁺ or Mg⁺⁺. Then

$$\begin{eqnarray} &&\chi ({\rm Mg}^+) \approx n_{{\rm Mg^+}} / n_{{\rm Mg^{++}}} = \frac{\alpha ({\rm Mg^+})}{c\sigma ({\rm Mg^+})} U^{-1}, \end{eqnarray} \tag{ 7 }$$

where the ionization parameter U is defined as

$$\begin{eqnarray} &&U \equiv \frac{Q}{4 \pi r^2 c n_e}, \end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray} &&\epsilon _{{\rm Mg+}} \approx \frac{\alpha _{{\rm Mg+}}}{c \sigma _{{\rm Mg+}}} \approx 4.7 \times 10^{-4}. \end{eqnarray} \tag{ 9 }$$

In the last expression, the recombination rate α_{Mg +}(T) ≈ 3.5 × 10⁻¹² cm³ s⁻¹ at T = 10⁴ K (Shull & van Steenberg 1982), and the photoionization cross section σ_{Mg +} ≈ 2.5 × 10⁻¹⁹ cm² (Verner et al. 1996). For 32016857, we estimate the ionizing photon luminosity Q ≈ 1.3 × 10⁵⁵ s⁻¹ from the Hα luminosity (see description in Table 1) and the relationship in Kennicutt (1998).

Using Equation (7), we can solve for n_H ≈ n_e

$$\begin{eqnarray} &&n_{{\rm H}} \approx \left[ \frac{n_{{\rm Mg^+}} Q}{4 \pi c \epsilon _{\rm Mg^+} \eta ({\rm Mg^+}), d({\rm Mg}) r^2} \right]^{1/2}. \end{eqnarray} \tag{ 10 }$$

At the location where τ₂₇₉₆ ≈ 1, the fiducial values give a H density

$$\begin{eqnarray} n_{{\rm H}} &=& 0.30\,{\rm cm}^{-3} \left(\frac{n_{{\rm Mg}^+}}{10^{-9}\,{\rm cm}^{-3} } \right)^{1/2} \left(\frac{11.4\,{\rm kpc}}{r_S} \right) \nonumber \\ &&\times \left(\frac{6.3 \times 10^{-2}}{d({\rm Mg})} \right)^{1/2} \left(\frac{3.8 \times 10^{-5}}{\eta ({\rm Mg})} \right)^{1/2} \left(\frac{Q}{10^{55}\,{\rm s}^{-1}} \right)^{1/2}.\nonumber\\ \end{eqnarray} \tag{ 11 }$$

The implied ionization parameter at the Sobolev point is

$$\begin{eqnarray} &&U(r_S) \approx 7.6 \times 10^{-2} \left(\frac{Q}{10^{55}\,{\rm s}^{-1}} \right) \left(\frac{0.30\,{\rm cm}^{-3}}{ n_e} \right)\left(\frac{11.4\,{\rm kpc}}{ r_S} \right)^2.\nonumber\\ \end{eqnarray} \tag{ 12 }$$

The resulting ionization fraction of Mg⁺,

$$\begin{eqnarray} &&\chi ({\rm Mg}^+) \approx 1.4 \times 10^{-3} \left(\frac{10^{55}\,{\rm s}^{-1}}{Q} \right) \left(\frac{ n_e}{0.3\,{\rm cm}^{-3}} \right) \left(\frac{ r_S}{11.4\,{\rm kpc}} \right)^2,\nonumber\\ \end{eqnarray} \tag{ 13 }$$

indicates most of the Mg in the warm phase of this outflow has been ionized to Mg⁺⁺. For comparison, inspection of Figure 5 in Murray et al. (2007) shows that χ(Mg⁺) lies between 0.1 and unity over broad range in gas density in ULIRG outflows due to the very low ionization parameter in dusty galaxies.

Using Cloudy version 13.00 (Ferland et al. 2013), we calculated the photoionization equilibrium of a constant density gas slab illuminated by a starburst and examined χ(Mg⁺) as a function of U. Over the range 8 × 10⁻⁴ < U < 0.08, a simple power law fit gives

$$\begin{eqnarray} &&\chi ({\rm Mg^+}) \approx n_{{\rm Mg^+}} / n_{{\rm Mg}} = \epsilon _{{\rm Mg+}} U^{-\alpha }, \end{eqnarray} \tag{ 14 }$$

with _{Mg +} = 4 × 10⁻⁴ and α = 0.94. Dropping the assumption introduced in Equation (7), Equation (10) is modified to

$$\begin{eqnarray} &&n_{{\rm H}} \approx \left[ \frac{n_{{\rm Mg^+}}}{ \eta ({\rm Mg^+}) d({\rm Mg}) \epsilon _{{\rm Mg+}}} \right] ^{1/(1+\alpha)} \left[ \frac{Q}{4 \pi c r^2} \right] ^{\alpha /(1+\alpha)}. \end{eqnarray} \tag{ 15 }$$

For these improved values of χ(Mg⁺) and α, Equation (11) becomes

$$\begin{eqnarray} n_{{\rm H}} &=& 0.15\,{\rm cm}^{-3} \left(\frac{n_{{\rm Mg}^+}}{10^{-9}\,{\rm cm}^{-3} } \right)^{0.52} \left(\frac{11.4\,{\rm kpc}}{r_S} \right)^{0.96} \nonumber \\ &&\times \left(\frac{6.3 \times 10^{-2}}{d({\rm Mg})} \right)^{0.52} \left(\frac{3.8 \times 10^{-5}}{\eta ({\rm Mg})} \right)^{0.52} \left(\frac{Q}{10^{55}\,{\rm s}^{-1}} \right)^{0.48}.\nonumber\\ \end{eqnarray} \tag{ 16 }$$

In the previous section, we argued that a sightline at impact parameter b ≈ 11.4 kpc through the outflow includes emission from r_S ≈ b up to r_S ≈ 12–18 kpc. For this range of radii and n_{Mg +} = 1.4 × 10⁻⁹ cm⁻³, our best estimate for the hydrogen density at the Sobolev radius becomes

$$\begin{eqnarray} &&n_{{\rm H}} \approx 0.12\hbox{--}0.18\,{\rm cm}^{-3}. \end{eqnarray} \tag{ 17 }$$

To facilitate comparison to TKRS 4389, we have adopted the same fiducial corrections for abundance and depletion as Rubin et al. (2011). In contrast, however, application of our argument for the ionization correction to TKRS 4389 will yield a significantly larger H density than the Rubin et al. (2011) estimate of n_H ≈ 0.003 cm⁻³ because they assumed χ(Mg⁺) ≈ 1. Using the ionic density n_{Mg +} = 2.6 × 10⁻⁹ cm⁻³ (from the previous section) in Equation (11), we estimate n_H = 0.28–0.67 cm⁻³ for r_S = 20–8 kpc. Over this same range of plausible values for r_S, our more accurate estimate from Equation (16) is n_H = 0.15–0.33 cm⁻³. We conclude that the gas density at the Sobolev points in the TKRS 4389 and 32016857 outflows are very similar.

Using the H gas density from the previous section and assuming 1.4 atomic mass units per hydrogen atom, the mass loss rate in the warm gas from Equation (1) becomes

$$\begin{eqnarray} \dot{M}(r) &=& 500\,{M_{\odot }\,{\rm yr}}^{-1} \left(\frac{\Omega }{\pi } \right) \left(\frac{f_c}{1} \right) \left(\frac{r}{11.4\,{\rm kpc}} \right)^2 \nonumber \\ &&\times \left(\frac{v}{230\,{\rm km\,s}^{-1}} \right) \left(\frac{n_{{\rm H}}}{0.3\,{\rm cm}^{-3}} \right). \end{eqnarray} \tag{ 18 }$$

We have used the observation that resonance lines are blueshifted 100 km s⁻¹ or more in 20% of star-forming galaxies at z ∼ 1 (Martin et al. 2012), and the interpretation that the outflows must subtend roughly 0.2 × 4π = 0.8π ≈ π sr on average. For 32016857, the mass flux in the (smooth) warm outflow is therefore approximately 330–500 M_☉ yr⁻¹, which is 4–6 times larger than the SFR of 80 M_☉ yr⁻¹ (for the Chabrier IMF). For TKRS 4389, we estimate a mass flux of 35 to 40 M_☉ yr⁻¹.

We can use the empirical limits on the Mg ii column density as a consistency check on the above argument. As can be seen in Figure 3, resonant emission fills in the Mg ii absorption troughs, so the absorption equivalent width must be estimated from a fitted model. Even the weaker transition of the Mg ii doublet is optically thick in any reasonable fit to the 32016857 spectrum; and, in Figure 3, it has an equivalent width of W₂₈₀₃ = 2.04 Å. We can use the linear relationship between W₂₈₀₃ and column density, valid in the optically thin limit, to place a lower limit on the ionic column density of N(Mg⁺) > 1.0 × 10¹⁴ cm⁻².

Substituting Equation (1) into Equation (2) defines the relationship between this column density and the mass-loss rate. For a constant velocity outflow, v(r) = V_Dop, it yields

$$\begin{eqnarray} &&\dot{M} \approx \Omega \bar{m}_{{\rm ion}} f_c N_{{\rm ion}} V_{{\rm Dop}} R_0. \end{eqnarray} \tag{ 19 }$$

The mass-loss rate estimated from absorption lines therefore scales linearly with the inner radius of the outflow, R₀, which physically represents a launch radius. Using our mass-loss rate estimate from the scattered emission and the lower limit on the column density, however, we obtain an upper limit on the maximum launch radius

$$\begin{eqnarray} R_0 &<& 2.0\,{\rm kpc} \left(\frac{\dot{M}}{500{M_\odot }}{\rm yr^{-1}} \right) \left(\frac{\pi }{\Omega } \right) \left(\frac{7 \times 10^{-16}\,{\rm g}}{\bar{m}} \right) \nonumber \\ &&\times \left(\frac{10^{14.5} cm^{-2}}{N_{{\rm Mg+}}} \right) \left(\frac{230 kms^{-1}}{V_{\rm Dop}} \right). \end{eqnarray} \tag{ 20 }$$

Dynamical models of outflows relate the launch radius to the disk scale height (Mac Low et al. 1989; Murray et al. 2011), suggesting values of R₀ ranging from ≈100 pc for thin gas disks to several kiloparsecs for dwarf galaxies and very turbulent systems. We conclude that the strength of the saturated absorption lines, while in no way providing a test of our large ionization correction, are fully consistent with the ionic density of Mg⁺ inferred from the scattered emission.

Up to this point, we have tacitly assumed a smooth outflow with f_c ≈ 1. Higher resolution spectroscopy of outflows, however, indicates that saturated, low-ionization lines are not always black; and this partial covering of the continuum light indicates that the warm outflows do not always have unity covering fraction (Martin & Bouché 2009). From a physical standpoint, interstellar gas is likely entrained in a hotter wind to create the outflow. In the Appendix of this paper, we outline simple physical arguments for the density and number of warm clouds in outflows. We find that the clumps have higher n_H than the smooth wind but reduce the overall mass-loss rate. The case of clouds confined by a hot wind produces higher gas densities in the outflow than do unconfined clumps. Both models suggest that clumps may lower the inferred mass-loss rate by up to a factor of 10. We conclude that the mass-loss rates in warm outflows from 32016857 and TKRS 4389 are at least 30–50 M_☉ yr⁻¹ and 35–40 M_☉ yr⁻¹, respectively. In this multi-phase model of the wind, we find the mass flux in the warm phase alone is comparable to the SFR.

We consider three models for the outflow, a smooth T ≈ 10⁴ K wind, an outflow consisting of unconfined T ≈ 10⁴ K clouds or clumps, and a two-phase hot (T_h ≈ 10⁸ K and smooth) together with a cold (T ≈ 10⁴ K) and clumpy outflow, in which the cold clouds are confined by thermal pressure of the hot gas. The hot gas in the last model is naturally provided by supernovae exploding in the galaxy, at a rate proportional to the SFR (about one supernova per century for a SFR of one solar mass per year).

A.1.1. The Smooth Wind Model

In a smooth wind the characteristic velocity spread is Δv = v_th and the characteristic length l is the Sobolev length

$$\begin{equation} l_{\rm eff} = l_{\rm Sob}\equiv {v_{{\rm th}}\over dv/dr}. \end{equation} \tag{ A8 }$$

The logarithmic derivative of the wind velocity

$$\begin{equation} \Gamma (r)\equiv {r\over v}{dv\over dr} \end{equation} \tag{ A9 }$$

is of the order of unity in the wind acceleration region, so

$$\begin{equation} l_{\rm Sob}=\Gamma ^{-1}\left({v_{\rm th}\over v}\right)r\ll r. \end{equation} \tag{ A10 }$$

A.1.2. Unconfined Clouds

Our second model assumes that the outflow consists of a large number of unconfined clouds. If the clouds are unconfined, e.g., if their gas pressure exceeds that of any hot outflow, then their size along the direction of acceleration g is

$$\begin{equation} l={c_s^2\over g}\approx \left({c_s\over v_c}\right)^2 r\approx 100\,{\rm pc }, \end{equation} \tag{ A11 }$$

where c_s ≈ 10 km s⁻¹ is the sound speed of the T ≈ 10⁴ K gas in the clouds, and v_c is the circular velocity of the host galaxy, which is of order the outflow velocity. The effective size of the cloud is

$$\begin{equation} l_{{\rm eff}}= \left({c_s\over \Delta v}\right) \left({c_s\over v} \right) \left({v_{\rm th}\over v}\right) r, \end{equation} \tag{ A12 }$$

where we approximate v_c ≈ v, the outflow velocity.

The velocity spread Δv ≈ v_th if there are no bulk flows in the clouds, in which case l_eff = l_Sob. In the more likely case that there are transverse bulk flows (expansions, in particular), then Δv, could be of order c_s. In that case,

$$\begin{equation} l_{{\rm eff}}\approx {c_s\over v}l_{\rm Sob}\approx 0.1l_{\rm Sob}. \end{equation} \tag{ A13 }$$

A.1.3. Confined Clouds

Our third model assumes that the cold gas consists of clouds confined by hot gas. With a SFR of ∼80 M_☉ yr⁻¹, and an expected supernova rate of nearly one per year, it is likely that the galaxy possesses a hot (T_h ∼ 10⁸ K) gas component. Since the sound speed of this hot gas exceeds the circular velocity of the galaxy by a large amount, it will flow out with a bulk velocity of a few times its sound speed, c_h ≈ 1000 km s⁻¹. This gas will act to pressurize the cold gas clouds, to accelerate them, and to disrupt them.

A priori, the cold clouds could have any size l_cl and a range of internal velocities Δv. However, we argue that there is a well defined lower limit to l_cl, as follows.

Both simulations (Klein et al. 1994) and experiments (Hansen et al. 2007) demonstrate that cold clouds embedded in such hot outflows are destroyed on a few tens of cloud crushing times. The cloud crushing time is defined as (Klein et al. 1994)

$$\begin{equation} \tau _{{\rm cc}}\equiv \left({n_c\over n_h}\right)^{1/2} {l_{{\rm cl}}\over v_s}, \end{equation} \tag{ A14 }$$

where $n_c=n_{{\rm H}}^{\tau =1}$ and n_h are the number densities of the cold and hot gas, and v_s ≈ v_h is the velocity of the shock in the hot flow.

The clouds must live long enough to reach R_{τ = 1} ≈ 11.4 kpc, which allows us to set a lower limit to l_cl:

$$\begin{equation} 30\tau _{{\rm cc}}\gtrsim \left(R_{\tau _{\rm 2796}=1}\over v \right), \end{equation} \tag{ A15 }$$

leading to a cloud length

$$\begin{equation} l_{{\rm cl}}\gtrsim \left({v_h\over 30 v}\right)^{4/3} \left({n_h\over n_{{\rm H}}^{\tau =1}({\rm Sob})}\right)^{2/3} \left({r\over l_{\rm Sob}}\right)^{4/3} \left({v_{\rm th}\over \Delta v}\right)^{1/3} l_{\rm Sob}. \end{equation} \tag{ A16 }$$

We have scaled the cloud survival time to 30 cloud crushing times. Scaling to a hot wind with a mass loss rate similar to the SFR, and assuming that Δv ∼ v_line,

$$\begin{equation} l_{{\rm cl}}\approx 170 \left({R\over 11.4\,{\rm kpc }}\right) \left({n_h\over 2\times 10^{-3}\,{\rm cm}^{-3}}\right)^{1/2} \,{\rm pc }. \end{equation} \tag{ A17 }$$

As with the unconfined cloud model, this length is similar to the Sobolev length associated with a smooth wind. The characteristic velocity spread Δv ∼ v_line; it cannot be larger than the observed linewidth. Then

$$\begin{equation} l_{{\rm eff}}\equiv \left({v_{\rm th}\over v}\right) l_{{\rm cl}} \approx 1.7\,{\rm pc }. \end{equation} \tag{ A18 }$$

If the gas being removed from the cloud is over-ionized (so that all the Mg ii ions are removed) before it is accelerated, Δv could be as small as v_th, but as we will see this would lead to a large mass loss rate in the cold component.

For a smooth outflow, the continuity equation

$$\begin{equation} \dot{M}_w = \Omega f_c \bar{m} r^2n_{{\rm H}} v \end{equation} \tag{ A19 }$$

gives Equation (11). Note that η_Mg, d_Mg, and $\epsilon _{\rm Mg^+}$ all enter to the one-half power, so that the estimate for the mass loss rate is only moderately sensitive to these quantities. Observations indicate, however, that the warm outflows are clumpy (Martin & Bouché 2009; Bordoloi et al. 2012). Taking f_c ≈ 1, where f_c describes the fraction of the outflow cone covered by clouds as seen from the galaxy, places an upper limit to the mass loss rate in warm (10⁴ K) gas for a fixed density, $\bar{m} n_{{\rm H}}$.

For illustration, we consider a bi-conical outflow of half-opening angle θ_bc such that the outflow subtends a solid angle $\Omega = \Omega _{{\rm bc}} \equiv 2 \pi \int _0^{\theta _{{\rm bc}}} \sin \theta d \theta$. In light of the limited observational constraints on the orientation of the outflows in 32016857 and TKRS 4389, the symmetry axis of the cone can be taken perpendicular to the observer's sightline to illustrate the impact of clumpiness on the density and mass-loss rate.

A.2.1. Clumpy Outflows

The mass of a single cloud is

$$\begin{equation} m_{{\rm cl}}={4\pi \over 3}l_{{\rm cl}}^3\bar{m} n_{{\rm H,cl}}. \end{equation} \tag{ A20 }$$

Over the velocity width of the emission line, Δv_el, clouds at the projected radius where τ₂₇₉₆(b) ≈ 1 have to reflect a large fraction of the continuum. To find the number of clouds at this impact parameter which have projected velocities consistent with the linewidth, consider an annulus of width Δb on the sky. Over the velocity width of a single cloud, Δv_cl, the clouds must cover a projected area 2θbΔb. In addition, the clouds have to cover the range of velocity seen in the emission line. Together these two constraints give a lower limit to the number of clumps or clouds given by

$$\begin{equation} N_{{\rm cl}}=2 f_v \left(\frac{\theta _{{\rm bc}}}{\pi } \right) \left({\Delta v_{{\rm el}} \over \Delta v_{{\rm cl}}}\right) \left({b\over l_{{\rm cl}}}\right) \left({\Delta b\over l_{{\rm cl}}}\right), \end{equation} \tag{ A21 }$$

where f_v is the fraction of the linewidth covered by clouds. The total mass in clouds is M_cl = N_clm_cl, or

$$\begin{equation} M_{{\rm cl}} = 2 \left(\frac{\theta _{{\rm bc}}}{\pi } \right) f_v \left({\Delta v_{{\rm el}}\over \Delta v_{{\rm cl}}}\right) \left({b\over l_{{\rm cl}}}\right) \left({\Delta b\over l_{{\rm cl}}}\right) \times {4\pi \over 3}l_{{\rm cl}}^3\bar{m} n_{{\rm H,cl}}. \end{equation} \tag{ A22 }$$

The mass loss rate is calculated using the crossing time,

$$\begin{equation} \tau _{\rm cross}= {\Delta b\over v_x(b)} = {\Delta b\over v \sin \theta } = {\Delta b\over v} \times {r\over b}. \end{equation} \tag{ A23 }$$

We find

$$\begin{equation} \dot{M}_{{\rm cl}} = \bar{m} r^2 v \times 4\theta _{{\rm bc}} f_v n_{\rm H,cl} \times {2\over 3} \times \left({l_{{\rm cl}}\over r}\right) \left({b\over r}\right)^2 \left({\Delta v_{{\rm el}}\over \Delta v_{{\rm cl}}}\right). \end{equation} \tag{ A24 }$$

Using Equation (A19), we scale the mass-loss rate in the clumpy outflow to the that of the smooth outflow,

$$\begin{eqnarray} \dot{M}_{{\rm cl}} &=& \left(\frac{4\theta _{{\rm bc}}}{\Omega _{{\rm bc}}} \right) \left(\frac{f_v}{f_c} \right) \times \left(\frac{n_{{\rm H,cl}}}{n_{{\rm H}}} \right) \nonumber\\ &&\times \left(\frac{2}{3} \right) \left(\frac{l_c}{r}\right) \left(\frac{b}{r}\right)^2 \left(\frac{\Delta v_{{\rm el}}}{\Delta v_{{\rm cl}}} \right) \dot{M}_w. \end{eqnarray} \tag{ A25 }$$

It is convenient to scale the cloud density to a smooth wind:

$$\begin{equation} n_{{\rm H,cl}} = n_{{\rm H}}({\rm Sob}) \left({l_{\rm Sob}\over l_{{\rm eff}}}\right)^{1/2}, \end{equation} \tag{ A26 }$$

where

$$\begin{equation} n_{{\rm H}}({\rm Sob})\equiv \left[ {n_\gamma n_{{\rm Mg\,\scriptsize{II}}}^{\tau =1}({\rm Sob})\over \eta _{{\rm Mg}}d_{{\rm Mg}}\epsilon _{\rm Mg^+}} \right]^{1/2}. \end{equation} \tag{ A27 }$$

For a clumpy outflow, the effective length scale for absorption over the linewidth contributed by a single clump is simply related to the properties of a cloud,

$$\begin{equation} l_{{\rm eff}} \approx \left(\frac{v_{{\rm th}}}{\Delta v} \right) l_{{\rm cl}}. \end{equation} \tag{ A28 }$$

The density of a clump or cloud in the wind will exceed that of the smooth wind by a factor

$$\begin{equation} n_{{\rm H,cl}} = n_{{\rm H}} \left({l_{\rm Sob}\over l_{{\rm cl}}}\right)^{1/2} \left({\Delta v_{{\rm cl}} \over v_{{\rm th}}}\right)^{1/2}. \end{equation} \tag{ A29 }$$

A.2.2. Comparison of Clumpy and Smooth Outflows

Using Equations (A25) and (A29),

$$\begin{eqnarray} \dot{M}_{{\rm cl}} &=& {2\over 3} \left(\frac{4\theta _{{\rm bc}}}{\Omega _{{\rm bc}}} \right) \left(\frac{f_v}{f_c} \right) \left({l_{{\rm Sob}} \over r}\right) \left({l_{{\rm cl}} \over l_{\rm Sob}}\right)^{1/2} \left({b \over r}\right)^2 \nonumber\\ &&\times \left({\Delta v_{{\rm el}} \over \Delta v_{{\rm cl}}}\right)^{1/2} \left({\Delta v_{{\rm el}} \over \Delta v_{{\rm th}}}\right)^{1/2} \dot{M}_w. \end{eqnarray} \tag{ A30 }$$

The square-root dependence of the mass loss rate on the cloud size l_cl indicates this estimate is fairly robust. To illustrate how the clouds affect the mass-loss estimate, we take r = (1.0–1.5)b ≈ b, 4θ_bc/Ω_bc ≈ 1, and f_v/f_c ≈ 1.

For an unconfined clumpy outflow, with l_cl ≈ 0.1l_Sob and Δv ≈ c_s, the mass loss rate

$$\begin{eqnarray} \dot{M}_{{\rm cl}} &\approx & 0.10 \left({\Delta v_{{\rm el}} \over 280}\right) \left({10 \over \Delta v_{{\rm cl}}}\right)^{1/2} \left({2 \over \Delta v_{{\rm th}}}\right)^{1/2} \left({l_{{\rm Sob}} \over 90}\right)^{1/2}\nonumber\\ &&\times \left({l_{{\rm cl}} \over 9}\right)^{1/2} \left({12 \over r}\right) \dot{M}_w \end{eqnarray} \tag{ A31 }$$

is about one-tenth the smooth wind rate. The mass-loss is also lower when we assume hot gas confines the clouds. For a confined clumpy wind, with l_cl ≈ l_Sob but Δv ≈ v_line, the mass loss rate is

$$\begin{eqnarray} \dot{M}_{{\rm cl}} &\approx & 0.06 \left({\Delta v_{{\rm el}} \over 280}\right) \left({280 \over \Delta v_{{\rm cl}}}\right)^{1/2} \left({2 \over \Delta v_{{\rm th}}}\right)^{1/2} \left({l_{{\rm Sob}} \over 90}\right)^{1/2} \nonumber\\ &&\times\left({l_{{\rm cl}} \over 90}\right)^{1/2} \left({12 \over r}\right) \dot{M}_w. \end{eqnarray} \tag{ A32 }$$

Allowing for very small cloud sizes l_cl, as might be the case in a hot-gas confined clumpy outflow, indicates that the mass loss rate can up to a factor of approximately 10 times lower than that estimated under the smooth outflow assumption.

Whether small clouds can survive the acceleration process remains open to some debate. The fiducial values of 9 and 90 pc used here lie at the low end of the size ranges recently estimated from photoionization modeling of circumgalactic gas clouds in both nearby galaxies (Stocke et al. 2013) and intermediate redshift absorption-line systems (Meiring et al. 2013). These calculations offer a physical argument nonetheless for the maximum amount by which the smooth outflow model may overestimate the mass-loss rate. They suggest that the mass-loss rate in the warm outflow could be just 10% of the smooth wind mass-loss rate. If such clouds are present in the wind, then the hot wind will of course contribute additional mass flux.