RESOLVING THE FAR-IR LINE DEFICIT: PHOTOELECTRIC HEATING AND FAR-IR LINE COOLING IN NGC 1097 AND NGC 4559^*

Using PACS on board Herschel we carried out far-IR spectral observations of NGC 1097 and NGC 4559 as part of the Kingfish SDP program. We report a detailed summary of observations in Tables 1 and 2. The integral field unit of PACS has a roughly 47'' × 47'' field of view composed of 25 94 × 94 pixels (Poglitsch et al. 2010). In Figure 1, we show footprints of the spectral observations of NGC 1097 and NGC 4559. These are aligned with Spitzer Infrared Spectrograph (IRS; Houck et al. 2004) long–low (LL) coverage. The long strips measure 240'' × 60'' and 290'' × 60'' for NGC 1097 and NGC 4559, respectively. In NGC 4559, observations are aligned with the major axis; whereas in NGC 1097 the LL strip is nearly orthogonal to the star-forming bar. The position angle of observations was determined by constraints in the pointing of the Spitzer telescope. In this work, we highlight resolved mapping observations obtained for three key emission lines ([C ii] 158 μm, [N ii] 122 μm, and [O i] 63 μm) in both Chop-Nod (CN) and Wavelength Switching²¹ (WS) modes (see Tables 1 and 2).

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All PACS spectral observations were reduced using the Herschel Interactive Processing Environment (HIPE) version 8.0 (Ott 2010). Reductions applied the standard spectral response functions, flat-field corrections, and flagged instrument artifacts and bad pixels (see Poglitsch et al. 2010). A dark was applied only to observations taken in WS mode, as CN observations remove the dark current via chopping. As inflight flux calibrations are not yet available for PACS spectral data, ground-based calibrations were applied to the data. These calibrations result in absolute uncertainties on the order of 30% and relative uncertainties of 10% and 20% in the blue and red grating orders, respectively.

WS data were offered as an alternative to chopping when a clean chop position was not available. In WS mode the grating is dithered by approximately half the FWHM of an unresolved line profile in an AABBBBAA pattern, where A is the reference grating position and B is the grating position of interest. The flux at the grating position of interest may be compared to the flux detected at the reference position to remove the baseline, rather than using a reference at the same grating position with a spatial displacement. Observations of an off position before and after each observation supplement the spectral dithering. Following the standard reduction procedures for PACS, as outlined by Poglitsch et al. (2010), we would subtract signal detected in the reference grating position from the grating position of interest. As the grating dither is small compared to the line width, this subtraction creates a differential line profile that must be modeled to reconstruct the data obtained in WS mode.

Fortunately, because of the exceptional thermal stability of PACS, we were able to process WS mode observations without modeling the differential line profile. Instead, observations were processed under the assumption that the baseline was stable for the duration of the observation. This meant that WS reductions were not dependent upon models of line profiles to measure the total flux. However, due to drifts and long-term transients, the astrophysical continuum was undetected. In essence, this reduction process was the same as that used for the unchopped mode that has since replaced WS in post-SDP observations.

The averages of the clean off observations obtained were subtracted from observations to correct for the thermal background contributed by Herschel. Subsequently, all spectra within a given spatial element were combined after offsetting each to the median continuum value. Approximately one-fifth of the measured ramps exhibited a highly unphysical continuum resulting from transients in the detector. These transients were strongly correlated with motions of the grating and of Herschel itself. Pixels which deviated from the mean spectrum by more than five standard deviations were masked out. Finally, spectral cubes with 5'' spatial pixels were created by projecting dithered raster positions of each spectral line together. In CN data, the two nod positions of the PACS array are slightly rotated on the sky with respect to one another. Therefore, the nods are treated as separate raster positions when projecting the observations.

To optimize the signal-to-noise ratio (S/N) and still take advantage of the spatial resolution, we employed 10'' apertures tiled every 5'' along the observed footprint. This aperture corresponds to ∼930 pc at the distance of NGC 1097 and ∼410 pc at the distance of NGC 4559. As apertures are not completely independent of their neighbors, there will be correlation between adjacent measurements. However, this approach avoids introducing bias into the sample by selecting apertures by eye. Furthermore, it fills out as large a parameter space as possible in the environment sampled for this study.

Some apertures exhibited a residual baseline after projection. To remove this, we fit a first-order polynomial to the spectral range outside one FWHM on either side of the line center as estimated from H i observations of NGC 1097 and NGC 4559 (Koribalski et al. 2004; Fischer & Tully 1981); this baseline fit to the spectrum was then subtracted. Subsequently, line fluxes for each aperture were determined from PACS observations using two methods. We measured the flux by directly integrating under the continuum-subtracted line. Additionally, we used Gaussian profiles fit to the spectrum extracted from each aperture to determine the flux (see Figure 2). For sources with detections above the 3σ level, we found that the flux measurements from both methods agreed within the relative uncertainties (20%) in NGC 4559. Lines from the central region of NGC 1097 are not well described by a Gaussian profile due to the large velocity dispersion of the star-forming ring (see Figure 2). Thus, the two flux determinations differed significantly in this spatial region. We have therefore adopted fluxes determined from direct integration. Outside the ring, both methods of integration over a line are in agreement.

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The PACS spectral observations of NGC 1097 and NGC 4559 are shown as projected line maps in Figures 3 and 4. Line maps of NGC 1097 have 3σ detection thresholds at a surface brightness of ∼0.04, 0.07, 0.01, and 0.07 × 10⁻⁷ W m⁻² sr⁻¹ in [C ii] CN, [C ii] WS, [N ii] CN, and [O i] CN lines, respectively. Line maps of NGC 4559 have 3σ detection thresholds at a surface brightness of ∼0.03, 0.05, 0.01, and 0.03 × 10⁻⁷ W m⁻² sr⁻¹ in [C ii] CN, [C ii] WS, [N ii] CN, and [O i] CN lines, respectively. While the [N ii] 122 μm and [O i] 63 μm lines were only detected in roughly 50% of PACS pixels, even short integrations of the [C ii] 158 μm line yielded detections in 80% of the PACS pixels. Indeed, only when observations were taken off the axis of NGC 1097 was [C ii] detection limited.

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Both galaxies were also observed by the Sings program (Kennicutt et al. 2003) with Spitzer-IRS (Houck et al. 2004). The central regions of NGC 1097 and NGC 4559 were targeted by the short–low (SL) module. The two extranuclear regions targeted in NGC 1097 were also observed with SL. In addition to SL observations, the long–low (LL) module targeted strips across each target galaxy (see Figure 1). There was no overlap of LL observations with the extranuclear positions in NGC 1097. For a more detailed description of the Spitzer data, including reduction techniques and a presentation of the central spectra of NGC 1097 and NGC 4559, we refer the reader to Smith et al. (2007b). For examples of the small aperture IRS spectra and maps of PAH emission we refer the reader to Beirão et al. (2012).

Spitzer-IRS spectra were extracted using CUBISM (Smith et al. 2007a). Broad spectral features and the mid-IR lines were fit and decomposed using PAHFIT (Smith et al. 2007b). We fixed the silicate absorption optical depth at zero as Smith et al. (2007b) found that negligible amounts of absorption from silicate dust were needed to reproduce the equivalent widths of lines in kpc scale regions of IRS spectral observations of NGC 1097 and NGC 4559. Only data that had a surface brightness larger than three times the root-mean-square (rms) noise were accepted as physically meaningful detections.

The PACS and IRS spectral observations are supplemented by imaging from 3 to 160 μm obtained as part of the Kingfish and Sings programs. These images are essential to the determination of the total infrared continuum emitted by dust, the far-IR color, and to aid in accounting for PAH emission where full spectral coverage is not available. The reduction of observations followed standard procedures, as detailed by Sandstrom et al. (2010) and Dale et al. (2005).

We convolve data to ensure that the comparison of data obtained at different resolutions is valid. This is particularly important as we are using a variety of instruments to cover a large wavelength range (3–160 μm). We are also interested in resolving the signatures of heating and cooling on small scales. These interests must be balanced as they lead us in diametrically opposite directions in the treatment of the data.

Convolving to a larger point-spread function (PSF) will degrade the resolution of our data, reducing the dynamic range of heating properties being sampled. This is further complicated by the fact that data near the edge of a map cannot be properly convolved as there is a hard edge past which no data is obtained. Therefore, at the edges of a map the flux is not conserved. While map edges lie well beyond the regions of interest in our imaging data, this is not the case for the spectral maps.

To test where we are limited in convolving the data, we have convolved masks of our IRS spectral maps (e.g., the value of each pixel recording data is set to one and pixels outside the footprint are set to zero) to the largest beam size used in this study (PACS PSF at 160 μm, FWHM = 1101) using sTinyTim (Krist 2002). We found this effect to be modest when convolving Spitzer-IRS LL spectra. Regions 22'' wide at the center of LL strips (see Figure 1) conserve approximately 98% of the flux, before dropping to 70% of the appropriate flux at the edge. However, the Spitzer-IRS SL spectra data suffer a substantial loss of flux; only 90% of the flux is conserved at the center of the spectral maps, dropping to 30%–50% at the edges. Furthermore, Pereira-Santaella et al. (2010, see Appendix A) show that the PSF of Spitzer-IRS spectral maps are not well reproduced by modeled telescope PSFs, suggesting that the generation of maps has significantly altered the PSF.

As a compromise between matching resolution and retaining reasonable spatial coverage without losing flux, we have convolved data with a beam smaller than the PACS 100 μm band (FWHM = 697) to this resolution and bootstrapped data with a beam larger than the PACS 100 μm band down to this resolution. Convolution kernels from Aniano et al. (2011) were used to convolve IRAC 3.6 μm (FWHM = 167), IRAC 8.0 μm (FWHM = 203), MIPS 24 μm (FWHM = 652), and PACS 70 μm (FWHM = 541) images to the resolution of the PACS 100 μm band (FWHM = 697). As the delivered PSFs of Spitzer-IRS are extremely complicated due to the mapping strategy employed by Sings (Pereira-Santaella et al. 2010), we have adopted a single Gaussian kernel appropriate for a central wavelength in each IRS module. Convolution of the [O i] 63 μm data (FWHM ∼ 53) was performed assuming a Gaussian PSF.

To overcome the resolution limits for PACS 160 μm (FWHM = 1101), we convolve PACS 70 μm observations to the resolution of PACS 160 μm. A representative 70 μm/160 μm ratio was measured in 12'' apertures centered on the associated spectral aperture. Similar to the approach in Smith et al. (2007b), we assume that this color ratio is the same in both the 12'' and 10'' apertures. This ratio is combined with the photometry from the PACS 70 μm observations that are convolved to the resolution of the PACS 100 μm band to derive a surface brightness for PACS 160 μm at this resolution. We calculate the TIR following the prescription of Draine & Li (2007),

$$\begin{equation} {\rm TIR} = 0.95\langle \nu S_\nu \rangle _8 + 1.15\langle \nu S_\nu \rangle _{24} + \langle \nu S_\nu \rangle _{70} + \langle \nu S_\nu \rangle _{160} \end{equation} \tag{ 1 }$$

using the matched 24 μm and 70 μm photometry combined with the bootstrapped PACS 160 μm, where the S_ν are the surface brightness values in the indicated bandpasses. While the [C ii] 158 μm line falls in the PACS 160 μm bandpass, contamination from this line is less than 2%, therefore we do not correct for its inclusion in the calculation of TIR.

Following our approach with the PACS 160 μm band described above, we have bootstrapped the [C ii] data to the resolution of the PACS 100 μm band. Here we assume that the ratio of [C ii]/νf_ν(70) does not change between 12'' apertures, which sample the full 160 μm beam, and the 10'' apertures. This ratio was combined with PACS 70 μm observations that were convolved to the resolution of the PACS 100 μm band to derive a [C ii] surface brightness in the 10'' apertures. Bootstrapping [C ii]/TIR and [C ii]/PAH in this manner alters their surface brightness by ≲5%. The most significant changes occur in the warmest regions of NGC 4559, which lie at the southwest edge of the [C ii] map next to a bright point source. The increased coverage of the 12'' aperture is not completely contained by the [C ii] map, thereby skewing the measured ratio of [C ii]/νf_ν(70).

To ensure that the conclusions of this study are not affected by convolving the data to the PACS 100 μm band rather than the 160 μm band, we have also convolved all our data to the resolution of the PACS 160 μm band. Using the coarser resolution and 12'' apertures to ensure that the PACS 160 μm beam is fully sampled, we reproduce the trends presented in this paper; however, the coarser resolution and larger apertures greatly reduce the number of reliable apertures.

Standard spectral observations with the PACS spectrometer require chopping between the target and a blank space of sky at a high frequency to remove the thermal signatures of the telescope and to correct for transients. The PACS chopper is limited to a maximum throw of 6'. When targeting many nearby galaxies, an offset of 6' is insufficient to avoid emission from the target galaxy in the offset field. NGC 1097 and NGC 4559 have minor axis diameters of 63 and 44, respectively. These sizes provide clean off positions within the 6' throw of the telescope chopper, given the orientation of the observations. Therefore, these galaxies provided an opportunity to verify that unchopped data can reproduce the line flux measurements from standard CN observations.

Line data acquired in WS mode have ∼40% less wavelength coverage than CN data due to the decreased number of grating steps taken in the WS observations. The increased spectral resolution of the blue channel results in insufficient wavelength coverage to ascertain an accurate placement of the continuum. Therefore, we restrict our comparison of these two modes, CN and WS, to data acquired from the red photoconductor array, which has lower spectral resolution. Spectral windows were centered on the nominal velocity of each galaxy. At the center of NGC 1097 the line-of-sight velocity differs by ∼500 km s⁻¹ on the northern and southern edges of the rotating inclined star-forming ring. This leads to wide double-peaked lines in many of the PACS spectral pixels. This rotational broadening of the line results in an increased uncertainty in the bright [C ii] line as even less continuum is available for accurate placement of the baseline.

$[{\rm C}\,\mathsc{ii}]$ 158 μm and [N ii] 122 μm fluxes measured from CN observations are plotted against matched observations in Figure 5. CN and WS data clearly agree within the ∼20% relative uncertainty of the flux calibrations. Observations of the [N ii] 122 μm line in NGC 1097 show the largest deviations from unity. This is most certainly due to the combination of the intrinsic faintness of the [N ii] line and the uncertainty of the continuum placement in WS mode in the broad lines at the center of NGC 1097.

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While both CN and WS modes prove to be sensitive to detections of emission lines, we adopt CN data when available due to the (1) increased wavelength coverage that enables a robust placement of the continuum and (2) the increased depth of the CN observations of [N ii] and [O i] which, in many cases, had integration times that were larger than the comparable WS observation by a factor of two or more. As both observation modes provide suitable line detections, we make use of the [C ii] WS data to extend the spatial coverage of both galaxies.

$[{\rm C}\mathsc{ii}]$ and [O i] luminosities could be diminished if the photons were absorbed by cool foreground gas (Luhman et al. 1998). Self-absorption has been observed in both the [O i] 63 μm line in Arp 220 (Fischer et al. 1997) and the [C ii] 158 μm line in observations of warm molecular gas in the Milky Way (Boreiko & Betz 1997). However, these occurrences of self-absorption were seen in highly obscured regions; Arp 220, a nearby ultraluminous infrared galaxy (ULIRG), has been shown to have unusually high optical depth (τ_100 μm ∼ 5; Rangwala et al. 2011); while the PDR observed by Boreiko & Betz (1997) was located on the far side of the molecular cloud, with a minimum possible optical depth of 0.6 in the [C ii] line.

Using the ratio of Hα/Hβ from Kennicutt et al. (2009) we find an average extinction, 〈A_v〉, of 1.40 and 0.50 for NGC 1097 and NGC 4559, respectively. Looking only at the 20'' × 20'' circumnuclear regions of each galaxy yields slightly higher amounts of extinction, 2.34 and 2.22 (Moustakas et al. 2010). However, these values are all considerably lower than the extinction seen in Arp 220 where self-absorbed [C ii] has been detected. It may also be noted that the extinction in the nucleus of NGC 4559 is only 0.14 mag lower than the extinction seen in the ring of NGC 1097. Despite this similarity, regions in the nucleus of NGC 4559 are not described by the warm dust temperatures seen in regions with a line deficit.

Cold foreground diffuse clouds with A_v ∼ 1.0 could produce an optical depth of ∼1.5 in the [C ii] line center. These clouds would have to contain cold C⁺ at the same velocity, within a few km s⁻¹, as the emitting cloud. A correlation in both velocity and position is unlikely between unassociated clouds. A chance alignment is expected to be more likely in edge-on systems, where more clouds could lie in the line of sight. Contrary to this expectation, the line deficit is primarily observed in NGC 1097, a roughly face-on galaxy, and not as robustly in NGC 4559 which is inclined 64° to our line of sight. This scenario becomes even less likely given that the position of these clouds must correlate with an increasing dust temperature.

Even so, we can determine an upper limit on the amount of C⁺ self-absorption present by using the silicate absorption features at 9.8 and 18 μm as a surrogate, since the mid-IR continuum absorbed by silicates originates in clouds that are likely to produce self-absorption. Scaling the silicate opacity will yield only an upper limit to C⁺ self-absorption as silicate absorption is insensitive to the velocity structure of the intervening clouds. The deconvolved [C ii] line width in most regions is narrower than the PACS instrumental profile at 157 μm (FWHM = 240 km s⁻¹); thus, line widths in our data are dominated by the instrumental profile. Observations of [C ii] emission in a galactic star-forming region and its associated PDR show widths of 5–10 km s⁻¹ using the Heterodyne Instrument for the Far Infrared (HIFI) on Herschel's (Dedes et al. 2010), which has a velocity resolution of 0.7 km s⁻¹. For narrow [C ii] emission, i.e., 4–10 km s⁻¹, the [C ii] line becomes optically thick at N(H) = 3.65–6.25 × 10²¹ cm⁻² (Russell et al. 1980), assuming N(C⁺)/N(H) = 1.6 × 10⁻⁴. Using the 10 μm silicate absorption profile from Li & Draine (2001), these column densities would produce a minimum silicate opacity, τ_9.7, of 0.135–0.270. We find marginal detections of silicate absorption in only 13 apertures where full Spitzer-IRS coverage exists. These regions, which are located on the western edge of the central ring of NGC 1097, yield 〈τ_9.7〉 = 0.06, with a maximum of τ_9.7 = 0.142.

While some self-absorption by C⁺ is possible, τ_9.7 yielded by fitting Spitzer-IRS spectra is not consistent with the existence of large amounts of self-absorption. We conclude that the existence of the far-IR line deficit in these galaxies is not attributable to self-absorption, which should introduce, at most, a drop in ([C ii] + [O i])/PAH by a factor of two. Nonetheless, we cannot eliminate the possibility that intervening molecular clouds may exist that increase the scatter we see in ([C ii] + [O i])/PAH and ([C ii] + [O i])/TIR.

$[{\rm C}\mathsc{ii}]$ emission in the local ISM will decrease after O and B stars have died, as photoelectrons are not ejected as frequently in the softer radiation field (e.g., Spaans et al. 1994), whereas the dust, including PAHs, continues to be heated. The existence of a soft radiation field in the warmest parts of a galaxy seems unlikely, e.g., a high νf_ν(70)/νf_ν(100) does not imply an absence of O and B stars. Indeed, recent observations of Paschen α emission in the central ring have shown that the very regions that exhibit a line deficit have a higher star formation rate than other parts of NGC 1097 (Hsieh et al. 2011). The elevated star formation and far-UV flux (Kuchinski et al. 2001) associated with the ring in NGC 1097, which is associated with many of the apertures that show a line deficit, lead us to reject a soft radiation field due to a dearth of massive stars.

The presence of large amounts of dust in the vicinity of photoionized gas could also lead to the creation of the line deficit (Luhman et al. 2003). Direct absorption of these photons by the dust also leads to an increase in TIR while decreasing the UV photons that escape and heat the neutral gas, thus leading to a net reduction in _PE. The models of Abel et al. (2009) have shown that the presence of dust in regions with a high ionization parameter removes Lyman continuum photons that would normally heat the gas. The absence of these photons leads to a reduction of far-IR fine-structure lines from both the ionized and neutral gas. However, with inclusion of radiation pressure, large grains may be compressed into an ionized shell, leading to an increase in the stellar photons that ionize H, compared to a uniform dust distribution, negating the ability of dust to cause a decrease in far-IR line emission (Draine 2011a).

Neither [N ii] 122 μm/TIR nor [Si ii] 35 μm/TIR show a trend with far-IR color, as illustrated in Figure 9. The lack of a deficit in the [N ii] line, which is only found in ionized gas, is particularly difficult to explain by invoking a dusty H ii region. If any trend is notable in [N ii]/TIR it is an increase in regions characterized by warm far-IR color. The different [N ii]/TIR plateaus may be related to the metallicity difference between NGC 1097 and NGC 4559, as secondary production of nitrogen may exist at the higher metallicity of NGC 1097 (Vila Costas & Edmunds 1993). Alternatively, this effect could also be the result of H ii regions in the ring of NGC 1097 filling a larger fraction of the ISM than other regions where [N ii] emission is detected.

Global studies of ULIRGs have indicated the presence of dusty regions and a deficit in the [N ii] 122 μm line (Fischer et al. 2010; Graciá-Carpio et al. 2011). We see no indication of this effect in resolved regions of NGC 1097 and NGC 4559. While we cannot rule out the applicability of dusty H ii regions to ULIRGs, we do not find evidence of elevated dust levels in the ionized gas within these two galaxies. However, Beirão et al. (2012) find $L_{{\rm FIR}}/M_{{\rm H_2}}$ ∼ 50 L_☉/M_☉ in NGC 1097. While this is below the ratio of L_☉/M_☉ where Graciá-Carpio et al. (2011) see a line deficit begin (80 L_☉/M_☉), it lies near this domain. If we plot data from Graciá-Carpio et al. (2011) with the ring of NGC 1097, we find that regions on the central ring of NGC 1097 lie at the lower edge of the scatter in their data and transition nicely into the deficit seen in galaxies with higher $L_{{\rm FIR}}/M_{{\rm H_2}}$.

To understand the far-IR line emission, we must also understand the conditions of the ISM that lead to far-IR line emission. We determine the average gas density, n_H, and radiation intensity, G₀ within each region of NGC 1097 and NGC 4559, assuming that all the emission originates from PDRs, by comparing [C ii], [O i], and TIR in each aperture to models of ionization from PDRs (Kaufman et al. 2006). Data are overlaid on plots of model [C ii]/[O i] versus ([C ii] + [O i])/TIR in Figure 10.

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According to PDR models, the gas which dominates the cooling in NGC 1097 and NGC 4559 lies at densities between 10^2.5 and 10³ cm⁻³. If the line deficit is caused by density effects, the [O i] line should become the dominant cooling line in the densest clouds. In contrast, as shown in Figure 8, rather than becoming the dominant line, [O i] exhibits a deficit congruent with the [C ii] deficit. This indicates that the observed line deficit is not a product of increased gas density.

The total integrated [C ii] emission along a line of sight includes contributions from ionized gas in addition to the PDRs. This additional [C ii] emission must be removed, as the models of Kaufman et al. (2006) only account for emission from neutral gas. The [C ii] contribution from ionized gas can be traced by emission from N⁺, which in the Galaxy is emitted primarily in the diffuse ionized phase (Bennett et al. 1994).

This correction is straightforward given measurements of [N ii] 205 μm, as this faint line has the same critical density as the [C ii] 158 μm line in ionized gas (∼ 45 cm⁻³) (Oberst et al. 2006). This results in a [C ii] 158 μm/[N ii] 205 μm ratio that is nearly constant (see Figure 11), varying by only a factor of 1.6 over three orders of magnitude in density (Oberst et al. 2006). Unfortunately, the faint [N ii] line at 205 μm is difficult to detect. The only observations of this line for these two galaxies cover the star-forming ring in NGC 1097.

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Beirão et al. (2012) employ the region in which [N ii] 205 μm emission is observed to determine that emission from diffuse ionized carbon is equal to ∼1.2 × [N ii] 122 μm. As we are interested in studying emission extended well beyond the brightest regions we must allow for this ratio to change. There are no observations of [N ii] 205 μm outside the nucleus of NGC 1097 that allow us to track these changes. Furthermore, no observations of [N ii] 205 μm exist to verify that this ratio is the same in NGC 4559, which has a lower star formation rate (Calzetti et al. 2010).

An alternative approach is to use the relatively stronger [N ii] 122 μm line as a means of tracing the diffuse ionized [C ii] emission (e.g., Kaufman et al. 1999; Malhotra et al. 2001, and references therein). The higher critical density of [N ii] 122 μm (∼300 cm⁻³) leads to a stronger dependence of [C ii] 158 μm/[N ii] 122 μm on density compared to [C ii] 158 μm/[N ii] 205 μm. This is shown in Figure 11 which plots the [C ii]/[N ii] line ratios as a function of electron density. Preferably, this density would be calculated from the ratio of [N ii] 122 μm/[N ii] 205 μm but, as discussed, the weaker line is unobserved for most of this sample. A substitute density indicator is available in the mid-IR ratio of [S iii] 18.71 μm/[S iii] 33.48 μm.

The [S iii] ratio is slightly temperature dependent and is most sensitive to densities in the range 100–10⁴ cm⁻³ (Snijders et al. 2007). We measure this ratio in our LL coverage of NGC 1097 and NGC 4559, when these lines are detected, and derive an electron density, n_e, using the models of Smith et al. (2009). We derive [C ii] 158 μm/[N ii] 122 μm appropriate for ionized gas as a function of n_e in order to scale the [N ii] emission appropriately. To determine this ratio, we used observed Galactic gas phase abundances of C (1.6×10⁻⁴ per hydrogen nucleus; Sofia et al. 2004) and N (7.5×10⁻⁵ per hydrogen nucleus; Meyer et al. 1997). For nitrogen we adopted the transition probabilities from Galavis et al. (1997) and collision strengths from Hudson & Bell (2005); while for carbon we use transition probabilities from Galavis et al. (1998) and collision strengths from Blum & Pradhan (1992).

The correction for diffused ionized gas using the [S iii] lines to trace the density is in agreement with the correction that would be adopted using the measurements of [C ii]/[N ii] 205 μm in the nucleus of NGC 1097, see Figure 11. We show the magnitude of this correction visually by plotting our data on models of PDR emission both with and without correcting for diffuse ionized gas in Figure 10. There is a significant shift in 20%–40% of the data points with a 3σ detection of [N ii] 122 μm, depending on assumptions in the low-density limit, when emission from ionized gas is removed. Given the uncertainties in correcting for contributions to [C ii] from ionized gas, the reality most likely lies between these representations of the UV field and the gas density we show in Figure 10.

When the [S iii] ratio is in the low-density limit, n_e ≲ 100 cm⁻³, the ratio of [C ii]/[N ii] 122 μm in ionized gas can vary between 1.24, at n_e of 100 cm⁻³, and 6.8, at n_e of 0.1 cm⁻³. Observations of the [N ii] 205 μm line (Beirão et al. 2012) indicate that at least some of the regions that are in the low-density limit have an n_e of ∼ 100 cm⁻³; therefore, in the nucleus of NGC 1097 we adopt the conservative scale factor of 1.24 appropriate for n_e = 100 cm⁻³. In regions outside the central star-forming ring of NGC 1097, the flux of the [N ii] 122 μm line is sufficiently weak relative to [C ii] that the effects of this uncertainty are negligible (e.g., <5%).

If observed PAH emission only receives contributions from the diffuse ionized gas, then '_PE should be calculated using only [C ii] emission that originates from PDRs. We show the fraction of [C ii] emission that emanates from PDRs as a function of the far-IR color in Figure 12. In addition to the correction adopted in this work, we also calculate the fraction of [C ii] that would originate in PDRs using the single value prescription from Malhotra et al. (2001). It is noteworthy that all regions show a relatively high PDR fraction, regardless of the far-IR color. Furthermore, the fraction of [C ii] emitted by the PDR decreases in the warmer regions, regardless of which correction for ionized gas is employed. An '_PE associated with pure PDR emission would thus exhibit an increased line deficit at warm far-IR colors, accentuating the sudden decrease in ([C ii] + [O i])/PAH.

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In addition to an ionized component and PDRs, the ISM also contains diffuse H i. Observations of the dust SED in these galaxies (G. Aniano et al. 2012, in preparation) are incompatible with the high G₀ we determine assuming that all the [C ii] emission and TIR originates in a PDR with an implied starlight intensity heating, 〈U〉, that is inconsistent, typically 10–50× lower than expected from G₀. In reality, some fraction of both TIR and the [C ii] line will emerge from the diffuse ISM. Attributing 70% of TIR and 30% of [C ii] surface brightness to cool diffuse gas would bring the G₀ from the PDR model and 〈U〉 from the dust SED into closer agreement, typically 〈U〉 1–10× lower than expected from G₀. This will simultaneously increase the n_H of the modeled PDR, as shown in Figure 13. Standard models of starlight heating indicate that this diffuse gas is too cool to emit [O i]. Decreasing the amount of [C ii] relative to [O i] emitted by the PDR could result in [C ii] no longer being the dominant coolant. However, the inclusion of other line coolants, e.g., [O i], does not eliminate the line deficit. It is noteworthy that models of starlight-heated diffuse gas also fail to produce the warm H₂ emission seen in some translucent clouds (Ingalls et al. 2011). H₂ emission lines are seen in our Spitzer-IRS spectra, indicating that the diffuse cloud population may also harbor hot material that supplements the [O i] emission, requiring non-standard heating processes that are not fully described by a PDR model.

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Assuming that diffuse gas is distributed in a relatively uniform manner throughout the ISM, correcting TIR, [C ii], and [O i] for its contributions would alter the absolute values of n_e and G₀, but would not change the general trend of points relative to each other, see Figure 13. PDR models are not able to reproduce all the spectral line and dust SED features we have observed. However, none of the corrections we have investigated would alter the fact that the ratio of G₀/n_e and 〈U〉 both increase in regions where the line deficit is observed. Contributions of line emission and TIR from diffuse gas, both ionized and cool, cannot produce the far-IR line deficit in NGC 1097 and NGC 4559.

Positively charged dust grains are inefficient at heating gas due to the greater potential well that must be overcome to eject additional electrons (Tielens & Hollenbach 1985). This decrease in gas heating is not accompanied by a strong decrease in the thermal dust emission that dominates TIR, nor does it cause the vibrational PAH emission to drop in strength (Draine & Li 2007). Therefore, an increase in grain ionization will lead to a deficit in line cooling relative to the thermal dust and PAH emission (Tielens et al. 1999).

Figure 10 shows that most of the gas in NGC 1097 and NGC 4559 lies at a density between 10^2.5 and 10³ cm⁻³. However, the local UV field, G₀, varies considerably, ranging from 50 to 1000, when the most conservative corrections for ionized gas are employed. We see the largest variation of UV heating in the ring of NGC 1097, while NGC 4559 displays remarkable uniformity in both n_H and G₀ across its disk. We compare ([C ii] + [O i])/PAH and ([C ii] + [O i])/TIR to G₀/n_H derived from models of PDRs and the mean dust-weighted starlight intensity, 〈U〉, derived from dust models of the IR SED (Draine & Li 2007), where the average value of 〈U〉 in each region was extracted from maps produced by G. Aniano et al. (2012, in preparation). Both gas and dust diagnostics clearly indicate that strong radiation fields are incident on the regions showing a line deficit. Indeed, G₀/n_H increases more than an order of magnitude between regions of low and high incident radiation, suggesting that grain charging could be important in these warm regions (Wolfire et al. 1990).

If the ionization of PAHs is indeed responsible for the observed line deficit, a significant variation in the 7.7–11.3 μm PAH band flux ratio should also be observed (Draine & Li 2007). The 7.7 and 11.3 μm PAH features are both observed using the SL module and have very similar PSFs, which are considerably smaller than the 12'' apertures we have used. As the convolution of Spitzer-IRS maps is very uncertain (Pereira-Santaella et al. 2010) we compare the 7.7–11.3 μm PAH band flux ratio measured from unconvolved spectra with the same ratio measured from convolved spectra. As expected, convolution of SL spectra decreases the spread seen in this PAH band ratio. The unconvolved spectra clearly show that more ionized PAHs are found in regions where the line-to-PAH deficit is strongest.

In Figure 14, we plot convolved [C ii]/PAH as a function of the ratio of the unconvolved 7.7 μm/11.3 μm PAH bands. Regions that show a far-IR fine-structure line deficit relative to total PAH emission do indeed exhibit a modestly greater than average 7.7 μm/11.3 μm ratio, suggesting that PAH ionization fraction is higher in regions with a line deficit. This is consistent with an elevated G₀/n_H seen in these regions.

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The shift seen in the 7.7–11.3 μm PAH band flux ratio could also be indicative of a change in the grain size distribution. Small PAHs are more efficient at producing photoelectrons, compared to the larger PAH grains, as they tend to stay neutral rather than becoming charged (Bakes & Tielens 1994). This has been confirmed via Galactic observations of the Horsehead nebula that show a smaller PAH/VSG (very small grain) ratio in the warm gas compared to the diffuse ISM (Compiègne et al. 2008). However, we do not detect a correlation between either the 7.7 μm/17 μm nor the 11.3 μm/17 μm PAH band ratio and ([C ii] + [O i])/PAH. If grain size was the driving factor in the suppression of ([C ii] + [O i])/PAH, we should detect a significant trend with both of these ratios as the 17 μm complex is attributed to the largest PAHs. Furthermore, we find that ([C ii] + [O i])/PAH does not correlate with the 6.2/7.7 μm band ratio, which has traditionally been associated with grain size (Draine & Li 2001).