PHYSICAL CONDITIONS IN THE LOW-IONIZATION COMPONENT OF STARBURST OUTFLOWS: THE SHAPE OF NEAR-ULTRAVIOLET AND OPTICAL ABSORPTION-LINE TROUGHS IN KECK SPECTRA OF ULIRGs

Longslit spectra were obtained with LRIS (Oke et al. 1995) and LRISb (McCarthy et al. 1998) on 2004 January 26, 2004 March 16-17, 2007 October 6, and 2007 November 1. Clouds and high humidity limited the exposure times on all of these nights. We present the five highest quality spectra obtained from the combined data.

The position angle of the longslit was selected to cover both nuclei of FSC 1009+47 and FSC 2349+24. The atmospheric dispersion compensator, which was installed on LRIS in spring 2007, was used for the 2007 observations. Care was taken with the 2004 observations to observe the targets when the slit PA was near the parallactic angle.

The dual beam spectrograph was configured with the D560 dichroic, a 1200-l grism blazed at 3400 Å in the blue arm (LRISb) and a 1200-l grating blazed at 7500 Å in the red arm. The grating tilt for each target was tuned to cover the Hα + [N ii] emission lines as well as the Na i absorption and He i 5876 emission line. The blue spectra cover the Mg ii 2796, 2803 doublet, the Mg i 2853 line, and the Fe ii 2587, 2600 doublet. The slit width was chosen based on the atmospheric seeing. The resolution (for a source filling the slit) ranged from 110 km s⁻¹ on the 2004 March run to 160 km s⁻¹ on the 2007 November run and is listed in Table 2. The size (FWHM) of the galaxies along the slit exceeded the slit width.

Fixed-pattern noise was removed using software scripts that called IRAF tasks.⁴ The blue spectra were flatfielded with twilight sky frames, normalized by the sky spectrum. The red spectra were flatfielded using an internal (spectrosopic) exposure of a quartz lamp. A dispersion solution was fitted to the vacuum wavelengths of the arc lamp lines as a function of detector coordinate. The root mean square error in the dispersion solution for the blue (red) spectra was 0.05 (0.07) Å. Application of a small, additive shift, up to a couple tenths of an angstrom, registered the wavelengths of night sky emission lines with their values in a telluric-line spectrum (Hanuschik 2003) smoothed to our spectral resolution and transformed to vacuum wavelengths (using the Edlen formula). We attribute these corrections to shifts in the dispersion solution with airmass and rotator angle. The correction to the Local Standard of Rest (LSR) was computed for each observation using the IRAF task RVCORRECT. All the offsets were less than 30 km s⁻¹. Since we are only concerned with relative velocites, the corrections to LSR were not applied to the data.

We rectified the two-dimensional spectral images, using the dispersion solution and traces of a standard star stepped along the longslit, and then extracted an integrated galaxy spectrum for each target. These spectra have SNR ∼ 5 − 10 per pixel as shown in Table 2. Additional, lower quality spectra were extracted for the distinct continuum sources within FSC 1009+47, FSC 1407+05, and FSC 2349+24. After substraction of the median sky intensity at each wavelength, significant residuals from strong night-sky emission lines remained in the red spectra near the Na i line in FSC 1009+47, 1407+05, and 1630+15. Variance spectra were extracted for each target prior to sky subtraction and flux calibration. The variance vectors were later scaled by a multiplicative factor that made the uncertainties in the intensity consistent with the measured standard deviation in the extracted, target spectra.

Observations of multiple, standard stars determined the relative sensitivity with wavelength. The data were flux calibrated using this sensitivity function and then normalized by a fitted continuum. The error in continuum placement is neligible around Na i, Mg i, and Mg ii; but the blending of stellar absorption lines washes out the true continuum level at shorter wavelengths. To identify bandpasses near Fe ii 2587, 2600 that likely reach the true continuum level, we compared high-resolution, synthesized spectra models of stellar populations to copies smoothed to 100 km s⁻¹ resolution. We fitted a first or second order cubic-spline through these bandpasses as well as the broad bandpasses near the Mg i and Mg ii lines. The resulting error in the continuum level near Fe ii depends on the star formation history. The severe blanketing in older bursts requires actual fitting of reddened, model spectra. This level of sophistication was unneccessary, however, because these near-UV spectra are bluer than the t > 100 Myr burst models; and we confidently rule out old, burst models for the continuum. The population synthesis models indicate either continuous star formation or a burst within the past 100 Myr. Repeated fitting trials indicate the uncertainty in the continuum level around Fe ii was 1 to 5%.

For a typical line width of 470 km s⁻¹ FWHM, we detect a rest-frame equivalent width W_r(5σ) ≈ 1.52 Å(5/SNR_pix)(Δv/470 km s⁻¹)^0.5 in the red spectra at the 5σ significance level. A typical 5σ sensitivity limit for the blue spectra is W_r(5σ) ≈ 0.92 Å(5/SNR_pix)(Δv/470 km s⁻¹)^0.5.

We measured redshifts from recombination lines of H and He and forbidden lines of singly ionized N and S. These lines all fall in our red spectra. The dispersion solution ties both the blue and the red spectra to the vacuum wavelengths of night sky emission lines. These redshifts agree with the previous measurements of Kim & Sanders (1998). Our independent check is important because that work used the Na i absorption line in the redshift estimate; and we will show that the Na i kinematics differ significantly from that of the recombination lines.

The sensitivity of the seven metal lines to outflowing gas depends on the relative abundances of the elements, the oscillator strengths of the transitions, the dust depletion, and the ionization corrections. In the limit of comparable ionization corrections for all species, the optical depth in the Mg i 2853 line would be the largest for solar-abundance ratios. The Mg ii lines have lower oscillator strengths, and the cosmic abudances of Fe and Na are lower. Our typical detection limit for Mg i corresponds to a minimum hydrogen column density,

$$\begin{eqnarray} N(HI) &=& 1.62 \times 10^{17}\, {\rm cm}^{-2} (N(Mg)/N(MgI)) (5/{\it SNR})\nonumber\\ &&\times\, \frac{1.25}{1+z}(\Delta v / 470\, {\rm km\, s}^{-1})^{0.5}. \end{eqnarray} \tag{ 1 }$$

Under the same conditions, the Na i 5898 line optical depth would be slightly lower than the weakest Fe ii line, but whether either transition is optically thin depends on the relative ionization correction, where

$$\begin{eqnarray} &&\frac{\tau _0(\hbox{{\rm Fe}\,{\sc ii} }2587)}{\tau _0(\hbox{{\rm Na}\,{\sc i} }5898)} = 1.41 \frac{\chi (\hbox{{\rm Fe}\,{\sc ii}})}{\chi (\hbox{{\rm Na}\,{\sc i}})}. \end{eqnarray} \tag{ 2 }$$

The Na i 5898 line can, in principle, provide the best measurements of an ionic column density in the outflow, but interpretation is complicated by the blending with the Na i 5892 transition, the ionization correction, and depletion of Na by dust grains. The Fe ii transitions with the lowest oscillator strengths lie blueward of the atmospheric cut-off for nearby galaxies, and their measurement will likely lead to the best estimates of mass column density.

When intervening absorption is studied in quasar spectra, or the Galactic halo studied in absorption against stellar spectra, the angular size of the absorbing gas clouds exceeds that of the continuum source. Gas between the observer and the light source attenuates the continuum by an amount proportional to the logarithm of the optical depth in any transition. For a doublet, we have

$$\begin{eqnarray} &&I_B(v) = I_0 e^{-\tau _B(v)} \\ &&I_R(v) = I_0 e^{-\tau _R(v)}. \end{eqnarray} \tag{ 3 }$$

For interstellar conditions, the relative optical depth between electronic transitions in a single ion from the ground state, i.e., of resonance lines, is τ_B/τ_R = (f_Bλ_B)/(f_Rλ_R). For the Na i and Mg ii doublets, the ratio of the oscillator strengths of the blue line, f_B, to the red line, f_R, is 2; and the wavelengths of the transitions are very similar. Substituion in Equation (3) indicates that the relative depth of the continuum-normalized absorption troughs must be I_B(v) = I²_R(v). These relations hold provided the clouds completely cover the continuum source over a velocity range comparable to (or greater than) the spectral resolution. In practice, when transitions with different f values saturate, both troughs appear black in noisy spectra.

Observations of Na i absorption troughs in ULIRGs cannot directly measure the relative intensities I_B(v) to I_R(v) due to the blending of the two absorption troughs. Previous modeling of the combined trough strongly suggests, however, that the assumption of complete continuum coverage is not met (Martin 2005, 2006; Rupke et al. 2005a, 2005b).

3.1.1. Description of Mg ii Absorption Troughs

3.1.2. Relative Shape of Mg i and Mg ii Absorption Troughs

In Figure 4, we scale the depth of each Mg ii absorption trough to facilitate comparison to the shape of the Mg i absorption trough. Based on our inspection of the data, we argue that the Mg ii and Mg i absorption troughs trace gas with the same kinematics.

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FSC0039 − 13: We reduced the depth of the Mg ii absorption trough by a factor of two by multiplying I_c(λ) − I(λ) by a factor 0.5 and adding the result to the normalized continuum, I_c(λ). The absorption troughs in Mg ii 2796, 2803 and Mg i 2853 are essentially indistinguishable after this scaling. The Mg ii absorption towards FSC0039 − 13 is detected at velocities ∼100 km s⁻¹ higher than the maximum velocities detected in Mg i. The SNR in the Mg i profile blueward of −600 km s⁻¹ is inadequate to definitively detect the weak absorption expected based on the shape of the Mg ii 2796 profile, so the Mg i observation is consistent with the presence of neutral Mg up to the same outflow velocity detected in Mg ii.

FSC1009+47: The scaled Mg ii 2796 and 2803 absorption trough is identical to that of Mg i blueward of −100 km s⁻¹ out to −600 km s⁻¹, where Mg ii 2803 blends with the red part of 2796. The high-velocity 2803 absorption makes the 2796 trough appear to be deeper than the 2803 trough between −100 km s⁻¹ and +150 km s⁻¹. The 2796 trough is detected to −780 km s⁻¹. The Mg i trough is cleanly detected to −450 km s⁻¹; but comparison to Mg ii 2803 suggests Mg i is plausibly present in the outflow to at least −700 km s⁻¹. After the depth of the Mg ii profile is reduced by a factor of 2.5, the Mg i profile is indistinguishable from the 2796 profile at high velocity and the 2803 profile at low velocity.

FSC1407+05: The Mg i and Mg ii 2796 lines in FSC1407+05 are both detected to velocities reaching −750 km s⁻¹. The Mg ii 2803 profile presents emission from 0 to +250 km s⁻¹. The Mg ii 2796 emission is diluted by the Mg ii 2803 absorption at the same wavelength. The addition of the 2796 emission to the Mg ii 2803 absorption trough elevates the latter between velocities −750 and −500 km s⁻¹. At intermediate velocities less affected by blending, −500 to 0 km s⁻¹, the Mg ii and Mg i profiles lie right on top of each other after reducing the depth of the Mg ii troughs by a factor of 2.0.

FSC1630+15: In FSC1630+15, the shape of the Mg ii and Mg i profiles are the same from 0 to −620 km s⁻¹ after scaling Mg ii by a factor of 2.0. Maximum absorption is offset to −400 km s⁻¹. The Mg ii 2796 absorption trough is detected to −825 km s⁻¹. The SNR in Mg i near these velocities is not adequate to detect the scaled Mg ii profile, so the velocity ranges of absorption Mg i and Mg ii are consistent with being the same.

FSC2349+24: The FSC2349+24 spectrum shows strong Mg ii emission. The 2796 emission fills in the bluest portion of the 2803 line profile. The 2796 trough reaches −650 km s⁻¹. We compare the Mg ii trough to Mg i without any scaling. The SNR is not adequte to detect the Mg i trough at outflow speeds greater than −500 km s⁻¹.

3.1.3. The Mg i Absorption Trough as a Template for the Blended Na i Trough

Since we detect these ULIRG outflows in neutral Na, the spectral energy distribution incident on the outflow likely presents a Lyman edge. (See the quantitative discussion in Section 2.3 of Murray et al. 2007). Stellar spectra have large intrinsic Lyman edges, so it follows that the starburst, rather than AGN, likely determines the Na ionization of the outflow. We expect a similar situation for Mg, which has an ionization potential just 2.5 eV higher than that of Na (7.6 eV versus 5.1 eV). The patchy distribution of dust in ULIRGs, however, leaves open the possibility that the morphological appearance of the stellar component in the near-UV differs considerably from that at 6000 Å. Absorption troughs present in these two bandpasses need not sample the same region of the outflow.

Before comparing the near-UV and optical trough shapes for the first time, it is useful to consider the relative optical depths in each transition should the absorption occur in the same clouds. Magnesium is 19 times more abundant than Na but only four times more depleted than Na.⁶ Singly ionized magnesium, Mg ii, is the dominant ion of Mg over a large range of density and temperature. In contrast, the Murray et al. (2007) ionization models predict a larger variation in χ(Na i) from object to object, allowing a wide range of possibilities for the relative optical depth of Mg i and Na i. For comparable Mg and Na ionization fractions, i.e., χ(Na i) ≈ χ(Mg ii) where χ(X⁺ⁿ) ≡ N(X⁺ⁿ)/N(X), the Na i 5892 optical depth and that of the weaker Na i 5898 line will be lower than the Mg i 2853 optical depth.

The blending of the Na i 5892, 98 lines complicates the direct, model-independent comparison. In Figure 5, we use the Mg i profile as a template for both the Na i 5892 (left column) and 5898 (right column) absorption troughs. In three of the ULIRGs, the depth of the Na i and Mg i absorption troughs are similar at minimum intensity. The two ULIRG spectra with strong He i 5877 emission present shallower troughs in Na i relative to Mg i, which we explain in part by emission filling of the Na i troughs. We take account of the He i 5877 emission filling in Section 3.3 and estimate comparable maximum outflow speeds in Na i, Mg i, and Mg ii. Surprisingly, the modeling in Section 3.3 demonstrates that the kinematics of the Na i absorbing gas are not distinghuishable from that in the low-ionization UV transitions.

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FSC0039 − 13: In the FSC 0039 − 13 spectrum, we detect Na i absorption up to −320 km s⁻¹. Emission filling from He i plausibly explains why the Mg i and Mg ii absorption troughs extend to higher velocity. The Mg i trough is nearly twice as deep as the bluer portion of the Na i trough (left panel) and three times deeper than the redder portion of the Na i trough (right panel). The He i emission contributes to the shallowness of the blue half of the Na i trough (relative to Mg i), but emission filling fails to expalin the low intensity of the red half of the Na i trough over 1000 km s⁻¹ from the He i line. The deepest part of the Mg i profile at +90 km s⁻¹ matches the minimum in the Na i trough. At the wavelength corresponding to the same Doppler shift in the Na i 5892 transition, we find another local minimum in the blended, absorption trough. We postpone discussion of the relative intensities of the Na i 5892 and 5898 absorption troughs to Section 3.3 because both transitions contribute to the absorption at minimum intensity.

FSC1009+47: In the FSC1009+47 spectrum, the Na i absorption trough extends from roughly the systemic velocity to about −450 km s⁻¹. At the coarse level of comparison allowed by the poor spectral SNR, the shapes of Na i 5892 and Mg i profiles match without any scaling. Better SNR at Mg ii shows the outflow reaches −800 km s⁻¹.

FSC1407+05: The Na i absorption is detected to −650 km s⁻¹, close to (within one resolution element of) the maximum absorption velocity detected in Mg i and Mg ii. The Na i and Mg i absorption trough shapes are very similar, with the latter being slightly deeper.

FSC1630+15: In FSC1630+15, the Mg i profile provides a good description of the Na i absorption trough with no scaling of the Na i 5892 or 5898 lines. Absorption is detected in both troughs to −600 km s⁻¹. The SNR's of the Na i and Mg i spectra do not allow detection up to the maximum velocity detected in Mg ii.

FSC2349+24: In FSC2349+24, He i emission fills part the blue-end, v < −250 km s⁻¹, of the Na i absorption trough. The Mg i and Na i troughs have similar intensity at other velocities.

3.1.4. Relative Strength of the Fe ii and Mg ii Absorption Troughs

In the spectral regions where the Mg ii 2796, 2803 absorption troughs can be directly compared, their intensities are nearly equal, I_B ≈ I_R. Since the galaxy (continuum) likely subtends a large angle relative to that of an individual shell fragment or cloud, we might expect variation in the properties of the absorbing material as a function of the spatial and/or velocity coordinates in the outflow. A number of methods have been developed to parameterize the relative geometry of the outflow and continuum source. We apply the most common method to the absorption troughs from FSC0039 − 13 and FSC1630+15, which show no sign of emission filling and little blending of the λ2796 and λ2803 lines. We then explain why two other parameterizations fail to describe the distribution of outflowing gas.

3.2.1. Pure Partial-covering Model

An approach used extensively for quasar winds (Arav et al. 2001; Arav et al. 2005; Arav et al. 2008) is to assume that a fraction 1 − C_f(v) of the area of the continuum source is free of absorption at velocity v; and the optical depth distribution in front of the covered part of the source is constant. For this parameterization, the line intensities are given by

$$\begin{equation} I_R(v) / I_0 = 1 - C_f(v) + C_f e^{-\tau (v)} \end{equation}\vspace*{-10pt} \tag{ 5 }$$

$$\begin{equation} I_B(v) / I_0= 1 - C_f(v) + C_f e^{-2\tau (v)}, \end{equation} \tag{ 6 }$$

where τ(v) is the optical depth at velocity v in the weaker line, here Mg ii λ2803. The coefficient in the argument of the exponential in Equation (6) depends on the relative oscillator strengths and wavelengths of the doublet transitions. We binned the blue and red profiles to the spectral resolution; and then computed the covering fraction

$$\begin{eqnarray} &&C(v) = \frac{I_R^2 - 2 I_R + 1}{I_B - 2 I_R + 1}, \end{eqnarray} \tag{ 7 }$$

where the continuum has been normalized to unity, I₀ = 1. At velocities where measurement errors produced unphysical line itensities, I_R < I_B or I²_R > I_B, we set the covering fraction to C(v) = 1 − I_R(v) or C(v) = 1.0, respectively. By substitution, the optical depth is

$$\begin{eqnarray} &&\tau (v) = ln \left(\frac{C(v)}{I_R(v) + C(v) - 1} \right). \end{eqnarray} \tag{ 8 }$$

Where statistical fluctuations result in I_R < I_B, measurement of the optical depth is not constrained. At velocities where I_R ≈ I_B, both transitions must be optically thick; and this method cannot discriminate an optical depth of 3 from 30 in galaxy spectra of typical quality. The limited spectral resolution and low SNR of these data leave open the possibility of substantial systematic errors in our best estimate τ(v). We will refer to this method as the pure partial covering scenario with velocity-dependent covering fraction.

Figure 6 shows the covering fraction of low-ionization absorption derived this way varies significantly across the trough. Our analysis necessarily neglects components narrower than 100 km s⁻¹, which may contribute to the total gas column but are unconstrained by our spectra. The covering fraction is near unity only where the trough is black. In the FSC0039 − 13 spectrum, this maximum coverage is near the systemic velocity regardless of whether we remove a symmetric, zero-velocity absorption component before computing C(v). In FSC1630+15, the velocity of the low-ionization gas with the highest covering fraction is between 300 and 400 km s⁻¹. The covering fraction falls steadily with increasing outflow speed in FSC0039 − 13, reaching 35% at 500 km s⁻¹. Similarly, away from the velocity where the absorption trough presents minimum intensity, C_f(v) declines in FSC1630+15. The residual intensity correlates with the derived covering fraction. We conclude that the shape of the absorption trough, I(v), is strongly influened by the velocity dependence of the covering fraction of low-ionization gas in the outflow.

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Figure 7 compares the Mg ii λ2803 optical depth, τ(v), to C_f(v). The low-ionization gas at velocity v ∼ −600 km s⁻¹ is apparently optically thick in the Mg ii transitions. At higher speeds, the blending of the blue wing of the Mg ii 2803 trough with the Mg ii 2796 line prevents a unique comparison of I_R(v) to I_B(v). The solution to Equation (8) for for FSC0039 − 13 and FSC1630+15 indicates that τ(v) increases by a factor of ∼2 and 1.5, respectively, as the covering fraction grows by a similar factor. Such a trend is physically plausible, even probable, but is not required by the data. The method only returns a lower limit on optical depth, τ ≳ 3. In our spectra, optical depths of 3, 30, and 300 yield essentially identical line profiles; in contrast, damping wings (not seen) would distinguish τ ≳ 10⁴. The lower limits on the optical depth, τ(v), determines the lower limit on the ionic column density in the ouflow at a given velocity. Applying Equation (1) from Arav et al. (2001) to the optical depth in the Mg ii λ2803 transition,

$$\begin{eqnarray} &&N [{\rm cm}^{-2}] = \frac{3.7679 \times 10^{14}}{\lambda _0[\AA] f} \int \tau (v) dv, \end{eqnarray} \tag{ 9 }$$

we estimate total ionic column densities of N(Mg ii)>6.4 × 10¹⁴ cm⁻² and >4.28 × 10¹⁴ cm⁻² toward FSC0039 − 13 and FSC1630+15, respectively.

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3.2.2. Inhomogeneous Absorption

The direct method of the previous section cannot be applied to the absorption troughs in which the doublet transitions blend together and/or the absorption trough blends with one or more emission lines. A quantitative description of these absorption troughs requires fitting the parameters of functions describing each spectral component.

A covering fraction, Doppler parameter, optical depth at line center, and Doppler shift describe the attenuation of the continuum by one component, or cloud, at every wavlength, $I_j(\lambda) / I_0 = 1 - C_j + C_j e^{-\tau _j(\lambda)}$. When the absorption trough is a blended doublet, the optical depth at a given wavelength may include significant contributions from each transition, and we add the doublet optical depths at that wavelength. At any particular velocity, the two transitions of the doublet must have the same C_f, b, and Doppler shift; and atomic physics fixes the ratio of their optical depths. At the systemic velocity of the He i 5877 and Mg ii emission lines, we fitted the amplitude of Gaussian intensity profile with velocity width matched to that measured for Hα.

The relative intensities of the Mg ii and Fe ii doublet troughs constrain these transitions to be optically thick, setting a firm lower limit on the optical depth, τ₀ ≳ 3, at line center. The blue skew of the absorption troughs requires at least two velocity components. When we fitted such models, however, we found a number of parameter degeneracies. We could describe the velocity width of each trough by adding additional velocity components, increasing the Doppler parameter, or further increasing the optical depth of individual components. For example, τ₀ can be made arbitrarily large by simply allowing the Doppler parameter b to shrink; and the product bτ₀ of a component can be made smaller (or larger) by adding (or removing) additional velocity components. Previous studies (Rupke et al. 2005a; Martin 2005, 2006) chose the minimum number of velocity components required to describe the Na i absorption troughs.

Simulations of winds show that a single sightline intersects multiple shell fragments (Fujita et al. 2009). And high resolution spectra of starburst galaxies resolve some broad absorption troughs into components (Schwartz & Martin 2004). With this picture in mind, we assume a sightline intersects a number of velocity components, where these structures partially overlap spatially. To allow for the possibility that some of the sightlines within the beam intersect different structures, we adopt the partial overlap model introduced by Rupke et al. (2005a). Each of the n successive velocity components attenuates the continuum (and the preceeding components) such that I(v) = ∏ⁿ_{j = 1}I_j(v). A Maxwellian distribution provides a reasonable description of the atomic velocities within each velocity component, so a Gaussian function models the optical depth distribution, $\tau (\lambda) = \tau _0 e^{-(\lambda - \lambda _0)^2/(\lambda _0 b/c)^2}$, of each component. The covering fraction is constant across a given component but varies from one component to the next. The Doppler parameter, b, describes the turbulent motion along the line-of-sight in a given gas cloud. For purposes of illustration, we fix the Doppler parameter at b ≡ 50 km s⁻¹ for all velocity components. For a typical minimum value of τ₀ ∼ 3, this choice yields a line width at half the trough depth around 100 km s⁻¹, consistent with the spectral resolution. We add velocity components until the fit statistic stops improving or a component becomes weaker than the detection limit. With this Doppler parameter, we need 6 velocity components to fit the Mg ii absorption trough in FSC 0039 − 13 and 5 velocity components for the other 4 galaxies.

Allowing τ₀ to vary among velocity components revealed no obvious trends with outflow velocity. The optical depth at the center of each component must be greater than three for the doublet transitions, but the probability distribution is very asymmetric with a significant tail reaching values ∼50 times higher than the minimum value. (See the Monte Carlo approach described by Sato et al. 2009 for describing uncertainties in optical depth.) We obtained an equally good fit statistic when we required a single value of τ₀ describe all the velocity components. This assumption of a velocity-independent optical depth is our most dubious prior, and we discuss how to improve upon it in Section 3.3.3. We adopt this approach for the purposes of comparing the velocity dependence of the gas covering fraction among objects and among transitions. For optically thick velocity components, the exact value of τ₀(v) has little effect on C_f(v). Our approach assumes the velocity depdendence of the covering fraction determines the trough shape.

We first fit C_f(v) and the minimum value of the optical depth to the Mg ii absorption troughs. We found that the same kinematic components described the Fe ii and Mg i absorption troughs well. In Section 3.3.1, we show that the minimum optical depth in the weak lines raises our best estimate of the minimum optical depth in Mg ii. We test the hypothesis these velocity components also describe the blended, Na i absorption troughs and compare covering fractions among species. Table 3 summarizes the fitted parameters. Figure 8 shows the resulting fits for Mg ii, Mg i, and Na i. We plot Fe ii separately in Figure 9.

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3.3.1. Fitted Optical Depth

For Mg ii, we can compare the ionic column inferred from the fitted partial-overlap model to those estimated in Section 3.2.1. With b ≡ 50 km s⁻¹, we construct the same ionic columns by setting τ₀(Mg ii2803) = 2.75, which implies an integrated optical depth across each velocity component of $\tau = \sqrt{\pi } c^{-1} \tau _0 b \lambda _0 = 2.3$ Å. The implied column densities,

$$\begin{equation} N[{\rm cm^{-2}}] = \frac{1.13 \times 10^{20} \tau _{\rm tot}[\AA]}{\lambda _o[\AA]^2 f}, \end{equation} \tag{ 10 }$$

add up to N(Mg ii) = 6.4 × 10¹⁴ cm⁻² for the 6 components describing FSC 0039 − 13. This model has the minimum τ₀ that comes close to describing the Mg ii absorption trough given our prior on b. The reduced chi-squared statistic improves significantly, from 1.425 to 1.143, when τ₀ is increased to 5. The fit statistic does not get much better as τ₀ is increased up to ∼50, so we consider τ₀ ≈ 5 to be our lower limit when the Mg ii absorption is considered independently. The pure partial-covering method also provided only a lower bound on τ₀(Mg ii) due to saturation and low spectral SNR.

Based on inspection of the Fe ii absorption troughs, we argue that the actual Mg ii 2803 optical depths are 2 to 3 times larger than the minimum required to fit the doublet. For a cosmic abundance ratio of gas-phase iron to magnesium, the optical depth in the stronger iron line is τ₀(Fe ii2600) ∼ 0.57τ₀(Mg ii2803)χ(Fe ii)/χ(Mg ii).⁷ For the weaker Fe ii line, τ₀(Fe ii2587) will always be 3.5 times lower than τ₀(Fe ii2600). It follows that τ₀(Fe ii2587) will be slightly less than unity if τ₀(Mg ii2803) = 5 and if our estimate for the relative ionization correction, χ(Fe ii) ≈ χ(Mg ii), is correct. This model predicts the Fe ii (2587) absorption trough is much shallower than the Fe ii (2600) trough as illustrated for FSC 0039 − 13 and FSC 1630+15 in panels a and e, respectively, of Figure 9.

For FSC 0039 − 13, this model spectrum is inconsistent with the shallow Fe ii (2587) trough, which requires τ₀(Fe ii2587) ∼ 3 or larger. As addressed further in Section 3.4, a difference this large in ionization fraction is unlikely given predictions from photoionization modeling. We conclude that the actual central optical depth in Mg ii 2803 is closer to 18 (than 5). Figure 9(d) shows that the resulting Mg ii fit is really indistinguishable from that presented in panel b. We find the fit to Fe ii 2587 in panel c preferable to that in panel a.

For FSC 1630+15, the Fe ii 2587 absorption trough is slightly deeper than the prediction from the minimum τ₀(Mg ii2803) model. Increasing τ₀(Mg ii2803) and τ₀(Fe ii2587) to 10 and 1.6 provides an acceptable fit; larger optical depths make the Fe ii 2587 trough deeper than observed in this case. We examined the UVBLUE spectra of synthesized stellar populations to determine if other lines might be blended with the Fe ii 2600 line. We identified resonance transitions from Mn ii, marked in Figure 3. The Mn ii 2594.499 transition does not cause the depression seen at slightly shorter wavelengths because the stronger Mn ii 2576.877 line is not detected. The SNR around the Fe ii 2587, 2600 lines is not adequate for further comparison in the other objects.

Our observation of a single Mg i transition does not directly constrain the Mg i 2853 optical depth. The optical depth in Mg i is

$$\begin{eqnarray} &&\tau _0(\hbox{{\rm Mg}\,{\sc i} }2853) = 6.1 \tau _0(\hbox{{\rm Mg}\,{\sc ii} }2803) \chi (\hbox{{\rm Mg}\,{\sc i}}) / \chi (\hbox{{\rm Mg}\,{\sc ii}}). \end{eqnarray} \tag{ 11 }$$

We argued that τ₀(Mg ii) reaches at least ∼10 − 18, so the neutral Mg fraction must be less than 1 to 2 percent to produce an optically thin Mg i 2853 trough. It seems more likely to us that Mg i 2853 is optically thick. The covering fraction then determines the shape of the Mg i absorption trough, I(v) ≈ 1 − C_f(v). Since covering fraction dictates both the Mg i and Mg ii absorption trough shape, they have a similar appearance, as shown quantitatively in the middle column of Figure 8. That the Mg ii troughs are deeper indicates that only a fraction of the cloud volume contains much neutral Mg.

The fitted models in Figure 8 assume equivalent kinematic components for Na i and Mg ii. For a cosmic abundance ratio, N(Mg)/N(Na) = 19.07, the Na i optical depth, τ₀(Na i5898) = 0.115τ₀(Mg ii2803)χ(Na i)/χ(Mg ii), will be lower than that of Mg ii 2803 for similar ionization fraction. In Section 3.4, we argue that χ(Na i)/χ(Mg ii) may be of order unity for soft spectral energy distributions (SED) with L^UV_ν/L^FIR_ν ∼ 10⁻⁹ but will vary among outflows, dropping to χ(Na i) = 0.001 for hard starburst SEDs.

When the weaker Na i line is optically thin, the fitted Na i models fall well short of the observed intensity in the redder half of the absorption trough. The models shown in Figure 8 have τ₀(5898) = 3 for 4 galaxies and 5 for FSC 2349+24. The ratio of these lower limits to those for τ₀(Mg ii2803) in the same objects is 0.3 or more. Since the photoionization models, and consideration of depletion factors, disfavor a situation with χ(Na i)/χ(Mg ii)>1, the moderate optical thickness required for Na i suggests τ₀(Mg ii2803) could easily be a factor ∼2 higher than that required by τ₀(Fe ii2587). As discussed in Section 3.1.1, however, the Na i and Mg ii absorption troughs could probe physically distinct regions of the outflow.

3.3.2. Constraints on Covering Fraction

The attraction of forcing the same kinematic components is that we can directly compare covering fraction of different species as a function of velocity, as summarized in Table 3. The fitted C_f values for the Fe ii velocity components come out similar to the values for Mg ii. The maximum covering fraction in low-ionization gas occurs at minimum intensity by construction, and we again associate the velocity of maximum covering fraction with the speed of swept-up shells at breakout. The covering fraction decreases towards higher outflow velocity.

Independent of any particular model, the shallow troughs of Mg i relative to Mg ii require a lower covering factor for the former at every velocity. The χ²_ν-fitting approach demonstrates that the shape of the Mg i absorption trough can be well described by the same velocity components as the Mg ii troughs and further quantifies the change in covering fraction. Figure 10 compares the fitted covering fractions for Mg ii and Mg i. The covering fraction for neutral gas remains roughly half that of the Mg ii at all velocities. Given the similar kinematics but constant offset in projected area, we suggest that the Mg i absorption originates in denser regions of larger structures traced by Mg ii.

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The fitted Na i covering fraction is similar to that fitted to Mg i for each velocity component. Sodium and magnesium have similar first ionization potentials, 5.1 eV for Na, and 7.6 eV for Mg. In the FSC0039 − 13 spectrum, emission filling could explain shallow Na i 5892 trough, relative to Mg i; but the emission is not broad enough to account for the shallow Na i 5898 trough. We suggested that the FSC 0039 − 13 sightline intersects more of the galactic, gas disk than our other observations; and the high inclination of this disk means we probe the swept-up shell where it has stalled in the disk. The extra attenuation at high inclination would likely produce larger than average differences between near-UV and optical continuum morhpology. High-resolution imaging in these bands may yield further insight into why the covering fraction of neutral, alkalai metals differs by almost a factor of two between the near-UV and optical sightlines. In more typical ULIRGs, the similar kinematics of Na i absorption and that of Mg i suggest the Mg i velocity components and C_f(v) might be applied as a template for deblending the Na i absorption trough into its Na i 5892, 5898 components.

3.3.3. Minimum Mass-loss Rate

The volume density of the low-ionization gas varies widely among outflow models. Figures 4 and 5 of Murray et al. (2007) illustrate the sensitivity of the Mg ii and Na i ionization fractions to the gas density. Since the saturated lines provide only lower limits on the column densities of Mg i and Mg ii, we cannot directly estimate the ionization fraction, thereby constraining the volume density. The null-detection of absorption from collisionally excited levels above the ground state provides a useful upper limit on the electron density, however. The most relevant excited transitions of Fe ii* are marked in Figure 3; see Korista et al. (2008) for a summary. We compute this upper limit and then discuss how it constrains ionization fraction in the outflow.

Above the critical density, the level populations approach their Boltzmann ratio, and we can easily calculate the relative strengths of absorption lines from different energy levels. The lowest energy level above the ground state, E = 385 cm⁻¹, has the lowest critical density. Our spectra cover the second strongest line from the multiplet with this lower energy level, Fe ii* 2612.6542.

In the FSC0039 − 13 spectrum, we expect Fe ii* 2612.6542 to be the strongest excited line. We place an upper limit on the observed equivalent width using the continuum SNR and assuming a line-width comparable to a spectral resolution element. The 5σ upper limit on the corresponding rest-frame equivalent width is 0.40 Å. For comparison, we also fitted a single-absorption-line model with b ≡ 50 km s⁻¹ and C_f ≡ 1 to the spectrum. At the systemic velocity, we find τ₀(Fe ii*2612) < 0.30, a value typical of this spectral bandpass. We obtain a more conservative limit if we repeat this procedure where the largest continuum deviation occurs at −850 km s⁻¹; and we find τ₀(Fe ii*2612) < 0.62. This more conservative limit corresponds to a column density, N(Fe ii*) < 6.26 × 10¹³ cm⁻² per component.

The optical depth in the individual velocity components of the weakest, detected resonance line is τ₀(Fe ii2587) ⩾ 3. Each of the 6 velocity components had b ≡ 50 km s⁻¹, so the lower limit on the column density in the ground state is N(Fe ii)>5.61 × 10¹⁴ cm⁻². Our measurements place a solid upper limit on the relative column densities in the first-excited and ground states, N(Fe ii*)/N(Fe ii) < 0.11, and likely less than 0.05. For densities well above the critical density this ratio would be ∼0.75 with a slight dependence on temperature. Using the level population calculations from Figure 3 of Korista et al. (2008), we conservatively limit the electron density to log n_e < 3.5 (or 3.4) for a temperature of (1 − 1.5) × 10⁴ K (or 5 × 10³ K), respectively. The stronger limit of N(Fe ii*)/N(Fe ii) < 0.05, which applies if the highest density gas is at low velocity, lowers log n_e to 3.1 (or 3.0), respectively, at T = (1 − 1.5) × 10⁴ K (or T = 500 K).

The gas density is important for understanding the relationship between the hot wind and the outflow observed in UV-optical absorption lines. Photoionization equilibrium likely sets the temperature of the low-ionization gas at about 10⁴ K. The density of the cool, low-ionization gas must be >50 cm⁻³ to be in pressure equilibrium with the hot wind, where P_h ≳ (10⁷K)(0.05 cm⁻³) ∼ 5 × 10⁵ K cm⁻³. Outflows accelerated by radiation pressure on dust grains do not require a hot wind at all (Murray et al. 2005), allowing the low-ionization gas to have much lower density. Our upper limit on the gas density does not challenge the multi-phase models.

Our result does eliminate the Murray et al. (2007) ionization models with n ∼ 6 × 10⁴ cm⁻³ or larger (see their Figures 4 and 5). We can confidently claim that Mg ii is the dominant ionization state of Mg. For any reasonable starburst spectral energy distribution (SED), χ(Mg ii) ⩾ 0.7; and the neutral fraction is less than 30%. The correction from the ionic columns of N(Mg ii) (and N(Fe ii)) to the elemental columns will be relatively minor. This knowledge of the Mg ionization balance suggests the Mg i 2853 optical depth significantly exceeds its minimum value of 3. Obtaining the same total column of Mg from the Mg i 2853 and Mg ii 2803 components requires

$$\begin{eqnarray} &&\tau _0(2853) = 6.1 \frac{\chi (\hbox{{\rm Mg}\,{\sc i}})}{\chi (\hbox{{\rm Mg}\,{\sc ii}})} \frac{C_f(\hbox{{\rm Mg}\,{\sc ii}})}{C_f(\hbox{{\rm Mg}\,{\sc i}})} \tau _0(2803), \end{eqnarray} \tag{ 12 }$$

or τ₀(2853) up to 5.2τ₀(2803) when χ(Mg i) = 0.3 and C_f(Mg i) = 0.5C_f(Mg ii).

Eliminating very high density rules out a high neutral fraction of Na. Interpolating between the curves in Figure 4 of Murray et al. (2007) to a density of 1000 cm⁻³, the Na i ionization fraction plumments from unity, when L_UV/L_IR ⩽ 10⁻⁹, to χ(Na i) ∼ 10⁻³ when the SED hardens to L_UV/L_IR ∼ 10⁻⁴. In contrast, only at very high gas density does the neutral Mg fraction change rapidly with spectral hardness. The ionization potential of Mg is 2.5 eV higher than that of Na..

Our spectrum of FSC2349+24 very clearly shows Mg ii emission. The emission line appears to be a bit redshifted, but we argue that the Mg ii absorption attenuates the blue side of the emission profile. The only other spectrum that definitely presents Mg ii is FSC1407+05. The Mg ii 2803 profile shows the emission from 0 to +250 km s⁻¹. The corresponding Mg ii 2796 line is less obvious because of Mg ii 2803 absorption at the same wavelength. We suggested that emission filling of the Mg ii trough explains the deeper absorption in the Fe ii trough relative to the Mg ii trough near systemic velocity.

Weiner et al. (2009) discovered Mg ii emission in a subset of z ∼ 1.4 galaxies with outflows and argued that the composite spectrum of the non-emission galaxies presented weak emission. They demonstated the presence of the emission by fitting the red-shifted portion of the Mg ii 2803 absorption trough with a symmetric absorption component at the systemic velocity. Removal of this component yielded a redshifted emission component in Mg ii 2796. This technique cannot be applied to four of five galaxies in our sample because their Mg ii absorption troughs present no absorption at v > 0 km s⁻¹. Removing a symmetric, zero-velocity component fitted to the redshifted absorption trough in FSC0039 − 13 reveals no significant emission excess in Mg ii 2796.

The origin of the Mg ii emission is not completely clear, but we adopt the hypothesis that the Mg ii lines are excited by recombination. The two objects presenting emission are classified as Sey 2 on the basis of their optical emission-line ratios, suggesting the ionizing spectrum is harder. The absence of extended Mg ii emission in the 2D spectra of FSC1407+05 and FSC2349+24 rules out a scattering origin from a galaxy-scale nebula.

The fitted profiles are shown in Figure 8 In FSC1009+47, the red side of the Fe ii troughs are marginally lower than those of Mg ii, hinting at a hidden emission component in the latter. However, inclusion of an emission component in the model for any of the other three galaxies does not improve the fit to the Mg ii absorption trough. In fact, any emission component at the systemic velocity significantly degrades the fit if the maximum intensity is more than 10% of the continuum level. We conclude that only two of the five ULIRGS present significant Mg ii emission. In contrast, an He i 5876 emission improves the model for the Na i absorption trough in all five galaxies. The line is strongest in FSC0039 − 13, which is the only object classified as an H ii galaxy in our sample.

In the popular "cool ULIRGs $\longrightarrow$ warm ULIRGs $\longrightarrow$ quasars" evolutionary scenario, as suggested by Sanders et al. (1988), one expects the AGN to provide the increase in bolometric luminosity. Advanced mergers in the 1 Jy sample tend to have higher luminosity (Veilleux et al. 2002). While our sample does contain rather luminous ULIRGs, their 25-60 μm colors, see Table 1, are all red enough to be classified as cool ULIRGS, i.e., f₂₅/f₆₀ < 0.2.

Following the morphological classification scheme outlined in Veilleux et al. (2002), the most advanced mergers we observed are FSC0039 − 13 and FSC1407+05. These are single nuclei systems with a compact morphology and little tidal structure. The dynamically younger merger, FSC1009+47, presents tidal tails but the nuclei have coalesced. The tails extend to 29 kpc in FSC1009+47 (Veilleux et al. 2002). Our second spectrum for FSC1009+47 is extracted 44 kpc south (along the slit) of the nucleus. Veilleux et al. (2002) classify this feature as a prominent knot in a tidal arm rather than a second nucleus because it is not detected in their K' image. Pre-merger sources with separated nuclei are rare in the 1 Jy sample, but we observed at least two such objects. FSC1630+15 is a close binary, separation 4.4 kpc (Veilleux et al. 2006); and FSC2349+24 is a wide binary, separation 14.5 kpc.

Among this small sample we found no evolutionary trends in low-ionization outflow properties. We compared the outflow properties along the sightlines to each nucleus in the double systems. The outflow was always seen in each spectrum.

The thermalized energy from supernova explosions drives a shock front through a galaxy, sweeping interstellar gas into a thin, radiating supershell (Tenorio-Tagle & Bodenheimer 1988; Shull 1993). While the growing shell remains smaller than the pressure scale height of the interstellar medium, it plows through an ambient medium of essentially uniform density; and the shell decelerates with velocity falling as v ∝ t^−2/5, or equivalently v ∝ R^−2/3, for a continuous injection of mechanical energy (Weaver et al. 1977). The shell mass grows linearly with the bubble volume at this stage, and the mass column through the shell grows linearly with radius

$$\begin{eqnarray} &&\bar{m} N(r) = 1/3 r \rho _0, \end{eqnarray} \tag{ 13 }$$

where ρ₀ is the average density of the ISM. During the supershell phase, the covering factor of the low-ionization gas will be unity provided the shell traps the ionization front.

Emission-line images of nearby, starburst galaxies show that supershells outgrow their host galaxies. We expect the shell to accelerate when the density gradient becomes steeper than ρ(r) ∝ r⁻² (McKee & Ostriker 1988), a highly unstable situation in which a dense shell pushes on more rarefied gas. Numerical simulations show that the shell breaks up due to hydrodynamic instabilities at a few pressure scale heights (Mac Low et al. 1989). The evolution of the shell fragments has received relatively little attention, but they are clearly one source of low-ionization gas that will absorb continuum emission. Additional sources of low-ionization gas include hydrodynamic instabilities induced by shear at the disk–wind interface (Heckman et al. 2000) and pre-existing, interstellar clouds over run by the supershell (Cooper et al. 2008).

A primary origin for the low-ionization outflows in shell fragments is appealing because it naturally explains the velocity offset of the Mg ii absorption troughs from the systemic velocity. Due to the high central concentration of gas in ULIRGs, the pressure scale-height is several times smaller than the value h_z ∼ 100 pc typical of normal galaxies; and the blowout radius R₀ ∼ 3h_z ∼ 200 pc. By the time of blowout, the mean speed of a shell has dropped to

$$\begin{eqnarray} v (R) &=& 164\, {\rm km\, s}^{-1} \left(\frac{L_w}{ 7.08 \times 10^{43}\, {\rm erg \,s}^{-1}} \right) ^{1/3} \left(\frac{10^3\, {\rm cm}^{-3}}{ n_H}\right) ^{1/3}\nonumber\\ &&\times\, \left(\frac{200 {\rm pc}}{R}\right)^{2/3}, \end{eqnarray} \tag{ 14 }$$

where the mass per H atom is $\bar{m} = 1.4 m_H$ in the ambient medium and L_w, the rate of mechanical energy injection, has been scaled to a SFR representative of ULIRGs, i.e., 100 M_☉ yr⁻¹.⁸ A shell velocity ∼200 km s⁻¹ can describe the Doppler shift of the deepest part of the Mg ii absorption troughs in Figure 1 b-1e. A sightline at high inclination (i.e., close to the plane of the gas disk) would intersect a lower velocity shell due to the higher average gas density in the plane. We appeal to this scenario as a plausible explanation for the kinematics of the Mg ii absorption troughs in FSC0039 − 13, which are deepest near the systemic velocity.

Numerical simulations of ULIRGs (e.g., Fujita et al. 2009) confirm that shell velocities decline to 200-300 km s⁻¹ by the time of blowout and suggest that, following blowout, the hot interior of the bubble accelerates outward creating a hot wind with terminal velocity $v_h \approx \sqrt{3} c_s \sim 940 T_7^{1/2}$ km s⁻¹. Fujita et al. did not follow the shell fragments very far into the halo; they simply assumed the clouds coast outwards on ballistic trajectories. Whether or not cosmic rays (Breitschwerdt 2008; Everett et al. 2008; Socrates et al. 2008), radiation pressure, and/or the ram pressure of the hot wind further accelerates the low-ionization gas remains an important question.

Murray et al. (2005) analytically modeled low-ionization outflows accelerated by the ram pressure of a hot wind and outflows accelerated by radiation pressure. It remains challenging to observationally distinguish not only these two types of momentum-driven outflows but also the energy-conserving, ballistic trajectories. The former accelerate quickly and then coast. The latter coast for a large distance before gradually decelerating. Factors favoring radiative-driving include an empirical correlation between outflow speed and escape velocity (Martin 2005), the high dust content of ULIRGs, and the high luminosity of ULIRGs. Reasonable objections include the small number of dwarf galaxies used in the outflow speed correlation, the difficulty of coupling radiative momentum to the gas in dust-poor, lower luminosity dwarf galaxies, and evidence for the presence of hot winds in ULIRGs (V. Sciortino & C. L. Martin 2010, in preparation).

For purposes of illustration, we consider acceleration by the ram pressure of a hot wind following blowout. The shell radius, R₀, and velocity, v₀, at blowout set the initial conditions at the start of the wind phase. In a spherical outflow geometry, the ram pressure drops as the inverse-square of the radial distance, so clouds accelerate over a relatively short spatial scale. The terminal velocity of any shell fragment depends on its column density, with the low columns reaching, at most, the hot wind velocity, v_c ≈ v_h (Martin 2005). The absorption troughs presented in this paper require the covering fraction of shell fragments to decrease with increasing velocity. This situation can be achieved by tuning the distribution of cloud column densities. The lowest column density fragments, which reach the highest terminal velocities, would need to be relatively rare, covering less area than higher column density clouds. Alternatively, the geometrical dilution of clouds offers a simpler way to achieve the desired result. We demonstrate this largely geometrical effect in Section 4.2. Further numerical work, beyond the scope of this paper, will be required to address the broader problem.

To illustrate the importance of geometrical dilution, let the locus of fractures in a shell define individual clouds. Suppose, for simplicity, these clouds all have the same column density and intitial velocity, equal to that of the shell just before blowout. If the area of a cloud does not change as it is moves outwards (think of a brick-like cloud), then the covering fraction of clouds decreases as C_f(R) = C_f(R₀)(R₀/R)².

Most shell fragments are unlikely self-gravitating structures, however; and we expect them to expand as they fly outwards due to the drop in ambient pressure. The sound crossing time in a cloud is likely short enough for the cloud to maintain pressure equilibrium with the hot wind, P_c ≈ P_h. The clouds expand adiabatically, so we can calculate their size at any outflow radius, R, and compare the cloud area to the that of a solid shell, thereby estimating the covering fraction, C_f(R).

Mass conservation in a steady-state, constant velocity hot wind requires the density to fall as ρ_h ∝ r⁻². For an isothermal hot wind, the resulting increase in the volume of low-ionization clouds is

$$\begin{eqnarray} &&V_c(R) = \left(\frac{R}{R_0} \right)^{2/\gamma _c} V_c(R_0), \end{eqnarray} \tag{ 15 }$$

where γ_c = 5/3 for a monatomic, ideal gas. For roughly spherical clouds, the increase in cloud volume is accompanied by an increase in cloud area, A_c ∝ V^2/3_c. Defining the covering fraction as

$$\begin{eqnarray} &&\frac{C_f(R)}{C_f(R_0)} = \frac{A_C(R)}{4\pi R^2} \frac{4\pi R_0^2}{A_C(R_0)}, \end{eqnarray} \tag{ 16 }$$

these relations yield a covering fraction,

$$\begin{eqnarray} &&\frac{C_f(R)}{C_f(R_0)} = \left(\frac{R}{R_0} \right)^{4/3\gamma _c -2}, \end{eqnarray} \tag{ 17 }$$

that falls as R^−1.2.

If the hot wind cools adiabatically, then T_h ∝ R^−4/3; and the pressure falls off faster with radius. The lower pressure allows the clouds to expand faster than in the isothermal case. Repeating the steps in the previous paragraph, we find

$$\begin{eqnarray} &&\frac{C_f(R)}{C_f(R_0)} = \left(\frac{R}{R_0} \right)^{4 \gamma _h / 3\gamma _c -2}. \end{eqnarray} \tag{ 18 }$$

Assuming γ = 5/3 for both the hot wind and the clouds, we find

$$\begin{eqnarray} &&\frac{C_f(R)}{C_f(R_0)} = \left(\frac{R}{R_0}\right)^{-2/3}. \end{eqnarray} \tag{ 19 }$$

The low-ionization clouds cannot expand quickly enough to keep up with the geometrical dilution inherent to spherical, outflow geometry. Their covering fraction must decrease with increasing outflow radius.

The absorption trough measurements require lower C_f at higher velocity. Geometrical dilution produces lower covering fractions at larger outflow radii. Combining these two results implies that the gas producing the highest velocity absorption resides at the largest distance from the starburst. While we find this argument illuminating, some of the underlying assumptions should be examined.

First, the outflow geometry is uncertain. It should be roughly spherical on spatial scales larger than the galaxy, but the outflow geometry may be better described as cylindrical at blowout. We know the absorbing gas lies close to the gaseous disk in some ULIRG outflows because the outflows present rotation in Na i (Martin 2006). In a version of the above model with cylindrical geometry, both the covering fraction and column density of "brick-like" clouds become independent of height above the disk. The decrease in covering fraction would require other physical effects such as cloud ionization, evaporation, or ablation to become important at the higher outflow velocities.

Second, blowout may yield a distribution of cloud column densities and velocities, something three-dimensional numerical simulations may soon be able to address. The fastest fragments—whether determined by blowout, acceleration by the hot wind, or acceleration by the starburst radiation—will reach the largest distances. Our models indicate the low-ionization cloud reach their maximum distance long after the starburst activity has ceased. It follows that if we observe such outflows during the starburst phase, the highest velocity absorption will come from material at the largest radii. Our underlying explanation for the shape of the absorption troughs would remain geometrical dilution; however, acceleration of the low-ionization outflow would not be required beyond blowout.

Comparison of the Mg ii, Mg i, and Na i absorption troughs indicates the highest velocity gas sometimes escapes detection in Na i and Mg i. For example, in FSC0039 − 13, we detect Mg i and Na i absorption to 300 km s⁻¹; but the Mg ii absorption demonstrates that the outflow persists to 500 km s⁻¹. In FSC1407+05, we measure terminal velocities of 350, 600, and 750 km s⁻¹ from Na i, Mg i, and Mg ii, respectively. We find our spectral signal-to-noise ratio insufficient to detect the higher velocity components of these outflows in Na i and Mg i.

The Mg ii lines allow more robust terminal velocity measurements for a number of reasons. First, the Mg ii 2796, 2803 transitions have a higher value of N(X)/N_Hf than does Na i 5892, 5898. Second, expected ionization corrections favor singly ionized Mg, Fe, and Na over their neutral species. Third, we find a larger cloud covering fraction for the singly ionized lines than for neutral lines.

We emphasize that the highest velocity absorption detected in Mg ii appears to be limited by covering fraction. At continuum SNR ∼ 10, the absorption trough blends with the continuum where C_f(v) ≲ 0.1, regardless of column density. The gas at the largest radii may well go undetected by typical absorption-line measurements. For example, if we assume unity covering fraction at R₀, then gas beyond R ∼ 3.2R₀ cannot be detected if C_f falls as R⁻². In the more realistic case, C_f ∝ R^−2/3, the covering fraction drops to 0.1 at 32R₀. For a launch radius R₀ ≈ 200 pc, our measurement would detect gas out ot 6.4 kpc. This distance exceeds the extent of the Hα filaments in many nearby starburst galaxies but remains considerably smaller than the recently detected soft, X-ray halos in ULIRGs (V. Sciortino & C. L. Martin 2010, in preparation). Spectroscopy of background light sources at projected separations of just a few tens of kpc from starburst and post-starburst galaxies should be more effective than line-of-sight studies for determining the spatial extent of the low-ionization outflow, but such studies need to account for the geometrical dilution of the clouds.