Regulating Star Formation in Nearby Dusty Galaxies: Low Photoelectric Efficiencies in the Most Compact Systems

The PAH line fluxes used in our analysis are reported in Stierwalt et al. (2013, 2014), and Inami et al. (2018). We take the PAH luminosity, L_PAH, to be the sum of features measured between ∼5 and 18 μm by Spitzer IRS. Specifically, L_PAH includes the 6.2, 7.7, 8.6, 11.3, and 17 μm PAH lines and all sub-features therein, as measured by CAFE which fits the line fluxes, continuum, and the extinction simultaneously (Marshall et al. 2007). On average, these five PAH lines account for 76% ± 9% of the total PAH luminosity measured by both the short-low (SL, 5.5–14.5 μm) and long-low (LL, 14 − 38 μm) slits in GOALS galaxies (Stierwalt et al. 2014). Because the SL and LL slits have different widths, tied to the changing point-spread function with wavelength, some highly resolved galaxies show a small jump between their SL and LL IRS spectra. This is discussed fully in Stierwalt et al. (2013), and we combine the SL and LL data using their scale factors derived from the wavelength overlap between the two slits. L_PAH is normalized to the SL slit. The 3.3 μm PAH luminosities (L_3.3) are also matched to the aperture extraction of IRS/SL data (see Inami et al. 2018 Section 3); however, L_3.3 represents a small component of L_PAH (5% ± 1.5%), and is therefore only used to investigate the properties of PAH line ratios. In general, the fraction of total PAH power measured through the SL slit compared to the LL slit does not correlate with galaxy distance, or other quantities we compare against in this work, namely Σ_IR. The sum of PAH features between 6 and 18 μm traces the bulk of the total power of PAH emission in GOALS.

From the sub-sample of GOALS galaxies with AKARI observations, 50 objects were selected as test cases for CAFE-NG (Advanced CAFE; P. Bonfini et. al. 2021, in preparation; Marshall et al. 2007), which simultaneously fits continuum and spectral features in the 0.6–35 μm range. These galaxies have full spectral model fits to the AKARI+Spitzer data, which explicitly include silicate absorption and 3.05 μm ice feature absorption which are fit independently. Joint fits to IR continuum and line measurements are important to get accurate line fluxes (e.g., Smith et al. 2007; Lai et al. 2020), and we use these new CAFE fits to estimate extinction-corrected 3.3 μm luminosities, L_3.3, for the full AKARI sample. In general, the measured PAH intensity depends on how the local continuum is estimated. Inami et al. (2018) adopt a spline technique (L_3.3,spline), which is known to under-estimate the extinction-corrected flux of PAH features between 6 and 17 μm by 30%–60% (Smith et al. 2007). From careful fits to the full PAH features (P. Bonfini et al. 2021, in preparation) we find that L_3.3,spline/L_3.3 = 0.29 ± 0.07. For the sample here we use this conversion and the results from Inami et al. (2018) to estimate L_3.3 for each source before comparing to the other PAH features.

Galaxies in GOALS vary from extended sources spatially resolved by Spitzer to unresolved point-like sources, which motivates a careful approach to aperture matching when comparing Sptizer and Herschel measurements. The effective area of the Spitzer/IRS SL slit is 37 × 95 (Stierwalt et al. 2013). To best match this area, we begin by using [C ii] and [O i] line fluxes measured through the central spaxel of the PACS/IFU with dimensions of ∼ 94 × 94 (See Section 3.2 in Díaz-Santos et al. 2017). Next, we project the effective SL slit and PACS central-spaxel aperture onto the Spitzer/IRAC 8 μm images, assuming that it traces the co-spatial PAH and far-IR line emission from star-forming regions in LIRGs, which is reasonable for the 2–7 kpc-scale observations in this work (Peeters et al. 2004; Alonso-Herrero et al. 2010; Pereira-Santaella et al. 2010; Hughes et al. 2016; Salgado et al. 2016). Smith et al. (2007) show the fractional power of PAHs to the IRAC 8 μm band to have a median value of 70% in normal star-forming galaxies, which increases only marginally when a more careful subtraction of stellar continuum is done using the IRAC 3–5 μm channels (e.g., Helou et al. 2004). We use the ratio of the 8 μm flux through each aperture to derive a correction, which is then used to scale down the PACS measurements to match the IRS aperture. The median of aperture corrections derived in this manner is 0.62 with upper and lower quartiles of 0.69 and 0.55 respectively. We note that aperture matching does not introduce any bias in the slope of trends we explore in this paper.

The [Si ii] 34.8 μm emission was measured in the Spitzer/IRS LL slit, using an effective extraction aperture of ∼106 × 366 (Stierwalt et al. 2013). To match measurements made through the IRS/SL slit, we scale the [Si ii] line flux down to the normalization of the SL spectrum using the same multiplicative factors applied to the 17 μm PAH fluxes (Stierwalt et al. 2013). As noted in Stierwalt et al. (2014), 15 LIRGs in GOALS have scale factors >2 due to extended structure in the LL slit not captured by the SL slit. To avoid complications, we do not include these sources in our analysis. In summary, we have matched the apertures between Herschel/PACS and Spitzer/IRS, and the measurements reported here are for the central regions (∼2–7 kpc) of GOALS systems unless explicitly stated otherwise.

The IR surface density is a more direct tracer of far-IR fine-structure line properties than color or L_IR(Díaz-Santos et al. 2013, 2017; Smith et al. 2017), both of which exhibit little to no influence on the ratio [C ii]/PAH (Figure 1). As Σ_IR should scale with the number density of massive star-forming regions in the beam, we expect Σ_IR to be more sensitive to statistical trends across the sample. The [C ii] deficit versus Σ_IR among star-forming LIRGs presented in Díaz-Santos et al. (2017) is re-created for comparison in Figure 2 (left). To test for trends in gas cooling over photoelectric heating, we calculate [C ii]/PAH as a function of Σ_IR, shown in Figure 2 (right), color-coded by $\mathrm{log}\,G/{n}_{{\rm{H}}}$ which scales linearly with IR surface density above $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ ∼ 10.7 (Díaz-Santos et al. 2017). Whereas the [C ii]/L_IR ratio exhibits a maximum deficit of ∼2 dex between $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ = 8–13, [C ii]/PAH falls by a factor of ∼0.5 dex over the same range. LIRGs at low Σ_IR exhibit [C ii]/L_IR and [C ii]/PAH comparable to those found in the spatially resolved regions of NGC 1097 (Beirão et al. 2012), and the normal star-forming galaxies in Helou et al. (2001).

Zoom In Zoom Out Reset image size

Following Díaz-Santos et al. (2017), we fit a second-order polynomial function to [C ii]/PAH of the form

$$\begin{eqnarray}&&\mathrm{log}{({L}_{\mathrm{line}}/{L}_{\mathrm{IR}})={a}_{0}+{a}_{1}\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}+{a}_{2}(\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}})}^{2}.\end{eqnarray} \tag{ 1 }$$

Note that the function is forced to a constant maximum at values of Σ_IR less than where the turnover occurs. While we do not include normal star-forming galaxies in the fit, a constant line-to-PAH ratio below the turnover yields the best agreement with observations of low-Σ_IR galaxies (e.g., Helou et al. 2001; Beirão et al. 2012; Croxall et al. 2012). The best-fit is shown as a black solid line in Figure 2 (right), the parameters for which are given in Table 1. While [C ii]/PAH is not constant, the magnitude of the drop with Σ_IR is significantly less.

Table 1. Best-fit Parameters to Line-to-PAH Ratios versus Σ_IR

Line	a0	a1	a2	1σ (dex)
[C ii]	−0.278	−0.124	0.00	0.08
fPDR × [C ii]	−2.378	0.242	−0.016	0.15
[O i]	−1.997	0.013	0.00	0.19
[Si ii]	−14.325	2.487	−0.120	0.19
PE a	−6.277	1.010	−0.050	0.12

Note. All line luminosities have been calculated from aperture-matched flux measurements as described in Section 3 unless otherwise noted. Fits are a function of log₁₀Σ_IR $\,=\,{\mathrm{log}}_{10}\,$(0.5L_IR $/\pi {R}_{\mathrm{eff},70\mu {\rm{m}}}^{2})$ in units of L_⊙ kpc⁻². The rightmost column corresponds to the 1σ scatter about the trend.

^a See Equation (2).

Download table as:  ASCIITypeset image

In addition to [C ii], [O i] and [Si ii] are important PDR cooling lines in terms of their overall contribution to the cooling budget (e.g., Rosenberg et al. 2015), and their strengths relative to the PAH emission are a diagnostic of PDR structure. Large [O i]/PAH where [C ii]/PAH is low could indicate greater gas densities, which would preferentially cool through the higher critical density [O i] line. In Figure 3, we show [C ii]/PAH, [O i]/PAH, and [Si ii]/PAH in separate panels for LIRGs and ULIRGs as a function of Σ_IR. Because we are comparing different fine-structure cooling lines here and would like to focus on the PDR emission alone, we subtract out the non-PDR component of the [C ii] emission using f_PDR (e.g., Díaz-Santos et al. 2017). At low IR surface densities, the line-to-PAH ratios are consistent with spatially resolved measurements of NGC 1097 (Beirão et al. 2012), representative of more normal star-forming conditions.

Zoom In Zoom Out Reset image size

[C ii]/PAH falls in the most compact sources at $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ ≳ 10. [O i]/PAH does not exhibit a turnover at high Σ_IR, but shows an increasingly larger scatter (higher than the other line ratios), mostly with a constant lower envelope. This could arise from varying degrees of line self-absorption, as [O i] originates from deeper regions within PDRs, and/or the presence of intervening (foreground) cold, neutral material. Indeed, IRASF1224-0624, and ICO860, the two galaxies >2σ below the [O i]/PAH trend both show signs of self-absorption in the PACS spectra and have some of the highest τ_9.7 optical depths observed in the sample; however, [O i] self-absorption is only observed in ∼5% of GOALS in total (12 objects; see Díaz-Santos et al. 2017) spanning $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ = 10–12.5, and is therefore unlikely to influence statistical trends in the sample over the full range in Σ_IR.

The photoelectric efficiency (_PE) is the fraction of energy that photoelectrons from PAHs transfer from UV photons into the gas relative to the total heating of dust. Observationally, _PE can be traced by the ratio of far-IR line emission to total PAH emission because photoelectrons from PAHs are one of the dominant heating mechanisms in PDRs, which cool predominantly via far-IR fine-structure line emission (e.g., Bakes & Tielens 1994; Malhotra et al. 2001; Croxall et al. 2012; Beirão et al. 2012). In practice, the cooling budget is dominated by [C ii], [O i], and [Si ii], and L_PAH ≫ (L_{[C ii]} + L_{[O i]} + L_{[Si ii]}). Assuming that the PAHs and ions are co-spatial such that the energy input into the gas by photoelectric heating powers the [C ii], [O i], and [Si ii] lines, an empirical tracer of the photoelectric efficiency is

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{PE}}\approx \displaystyle \frac{{L}_{[{\rm{C}}\mathrm{II}]}+{L}_{[{\rm{O}}{\rm{I}}]}+{L}_{[\mathrm{Si}\mathrm{II}]}}{{L}_{\mathrm{PAH}}}\end{eqnarray} \tag{ 2 }$$

where the relative contribution of far-IR lines to _PE can vary across the PDR (Kaufman et al. 2006). Nevertheless, combined observations of these features capture an average _PE of PDRs over a galaxy.

We measure the average _PE for 152 GOALS star-forming galaxies including five upper limits, and show the combined sum of neutral [C ii], [O i], and [Si ii] cooling over L_PAH in Figure 4. A Kendall's τ-test supports an anti-correlation between _PE and Σ_IR at ∼5σ (see Table 2). As shown in Figure 4, LIRGs at low Σ_IR exhibit fairly constant photoelectric efficiencies comparable to observations of normal star-forming galaxies (Beirão et al. 2012; Croxall et al. 2012). At high Σ_IR, the sum of all cooling lines relative to the PAHs shows a decreasing photoelectric efficiency, which suggests that the net cooling to photoelectric heating ratio is falling across the density and temperature structure of PDRs in compact objects. The relative contributions of [C ii], [O i], and [Si ii] to the cooling budget within PDRs (i.e., the numerator of Equation (2)) are on average 32%, 20%, and 48% respectively, and these values do not change significantly when splitting the sample above and below $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ = 10.7. Therefore, low photoelectric efficiencies are the product of overall less cooling in PDRs in all channels relative to PAH heating.

Zoom In Zoom Out Reset image size

Table 2. Correlation Coefficients between the Photoelectric Efficiency, as Defined by the Ratio of IR Cooling Lines to Mid-IR PAH Emission between 6 and 17 μm, and Other Quantities in GOALS Star-forming Galaxies

	τk	$\mathrm{log}p$	SNR [σa ]
LIR [L⊙]	$-{0.15}_{-0.2}^{-0.1}$	$-{2.21}_{-3.7}^{-1.1}$	$2.7{\,}_{1.7}^{3.7}$
Sν,63μm/Sν,158μm	${0.00}_{-0.09}^{0.08}$	$-{0.35}_{-0.91}^{-0.09}$	$0.8{\,}_{0.2}^{1.5}$
ΣIR [L⊙ kpc−2] b	$-{0.30}_{-0.4}^{-0.2}$	$-{7.1}_{-9.5}^{-4.9}$	$5.3{\,}_{4.4}^{6.3}$
G/nH [G0 cm−3] c	$-{0.31}_{-0.35}^{-0.26}$	$-{9.0}_{-11.6}^{-6.6}$	$6.1{\,}_{5.2}^{7.0}$
EW(6.2 μm) [μm]	${0.08}_{0.01}^{0.1}$	$-{0.8}_{-1.9}^{-0.2}$	$1.5{\,}_{0.4}^{2.5}$
fAGN,MIR	${0.08}_{0.03}^{0.1}$	$-{0.9}_{-2.1}^{-0.3}$	$1.6{\,}_{0.6}^{2.6}$
fPDR	${0.06}_{0.0}^{0.1}$	$-{0.6}_{-1.5}^{-0.1}$	$1.1{\,}_{0.3}^{2.2}$
L11.3 μm/L7.7 μm	${0.06}_{0.0}^{0.12}$	$-{0.6}_{-1.7}^{-0.2}$	$1.2{\,}_{0.4}^{2.3}$
L7.7 μm/L6.2 μm	$-{0.23}_{-0.3}^{-0.2}$	$-{4.9}_{-7.1}^{-3.1}$	$4.3{\,}_{3.3}^{5.4}$
L11.3 μm/L3.3 μm d	$-{\bf{0}}.{{\bf{37}}}_{-0.42}^{-0.32}$	$-{\bf{7}}.{{\bf{5}}}_{-9.5}^{-5.7}$	${\bf{5.5}}{\,}_{4.8}^{6.2}$

Notes. Kendall's τ correlation coefficients (τ_k ) and p-values were calculated using pymccorrelation (Privon et al. 2020), which implements bootstrap error estimation on τ_k and p with censored data (upper limits) as described in Curran (2014) and Isobe et al. (1986) respectively. We report 16%, 50%, and 84% percentiles for each quantity.

^a Assuming normally distributed posteriors. ^b See Figure 4. ^c See Figure 5. ^d For the subset of GOALS (U)LIRGs with AKARI 3.3 μm PAH detections. See Figure 7.

Download table as:  ASCIITypeset image

The anti-correlation between _PE and Σ_IR is robust against outliers, as demonstrated by the bootstrap fits on Figure 4. Nevertheless, there exist galaxies deviating from the trend by more than ±2σ that warrant closer inspection. At all IR surface densities, objects in this class collectively have higher mean and median f_AGN,MIR compared to the rest of the sample. Thus, the scatter above and below the best-fit trend may be partially due to sources with an excess of hot dust continuum emission in their mid-IR spectra. This may be due to a weak, deeply buried AGN, even in sources where L_IR is dominated by star formation, which in turn may change the ionization structure and the PAH properties (Voit 1992; Langer & Pineda 2015; Langer et al. 2017). Alternatively, these objects may harbor heavily obscured star-forming regions with high levels of line self-absorption and/or continuum opacities. This absorption should appear in the PACS line profiles, but only 12 GOALS sources show evidence for absorption in the [O i] line (Díaz-Santos et al. 2017). We note that the detection of these signatures, presumably due to compact (narrow-velocity) foreground clouds, may be limited by the coarse spectral resolution (∼85 km s⁻¹) of Herschel/PACS (Gerin et al. 2015).

The photoelectric efficiencies we measure represent averages over the local properties of PDRs in the central (∼2–7 kpc) star-forming regions of LIRGs, and the trend in Figure 4 suggests a departure from the typical ISM and PDR conditions at high Σ_IR. Indeed, galaxies in this domain with low _PE also have the largest ratios of G/n_H, which are also sensitive to the local physics of PDRs. We fit Equation (1) to the data, the parameters of which are given in Table 1, and find that the photoelectric efficiency turns over at $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ ∼ 10.5, close to where G/n_H scales linearly with log Σ_IR (Díaz-Santos et al. 2017). Models of heating and cooling in PDRs predict an anti-correlation between G/n_H and _PE as the size and charge of dust grains are modified by the stellar radiation field (e.g., Bakes & Tielens 1994; Tielens 2008), a trend which we recover in Figure 5 at high (∼6σ) significance (see Table 2). We find general agreement between modeled and measured _PE for T_gas = 50–1000 K, reasonable values for temperatures across the phase structure of PDRs (e.g., Tielens & Hollenbach 1985; Bakes & Tielens 1994; Draine & Bertoldi 1996), and the two sets of models are able to account for the range in _PE we measure in GOALS. The most extreme ULIRGs at high Σ_IR and low _PE also show the highest values of $\mathrm{log}(G/{n}_{{\rm{H}}})\gtrsim 0.25$, supporting a link between the strength of the radiation field and the heating efficiency within the PDR, which is mediated by the properties of PAH grains.

Zoom In Zoom Out Reset image size

The low photoelectric efficiencies we measure in galaxies with high Σ_IR (Figure 4) and high G/n_H (Figure 5) suggest a change in the thermal regulation of star-forming gas in compact LIRGs; however, a number of alternative physical conditions may conspire to produce the trends we observe with _PE.

The relative strengths of the far-IR cooling or PAH emission may be affected by any or all of the following. (1) Far-IR line fluxes may be suppressed by optically thick continuum absorption (e.g., Scoville et al. 2017a). In this case, the [O i]/L_IR deficit would have a steeper slope than [C ii]/L_IR (e.g., Malhotra et al. 2001), a trend that is not observed (Díaz-Santos et al. 2017). (2) Ionized gas from diffuse or H ii regions, and/or AGNs may contribute to the [C ii] and [Si ii] line fluxes owing to their ionization potentials being lower than that of neutral hydrogen. This is unlikely to drive trends in our sample, as the photoelectric efficiency does not correlate with the fraction of [C ii] emission from PDRs (f_PDR; Table 2). Moreover, the neutral PDR fraction of [Si ii] would have to be larger at low Σ_IR or smaller at high Σ_IR to maintain constant _PE, both of which are inconsistent with the increase in PDR area-filling factor with Σ_IR in LIRGs and ULIRGs (Díaz-Santos et al. 2017). (3) PAH molecules in galaxies with large G/n_H absorb and re-emit more energy per unit dust mass because of the large UV opacities of grains (Li & Draine 2001). If the PAHs and ions (C⁺, O, and Si⁺) were decoupled spatially, L_PAH could increase with Σ_IR without a corresponding increase in (L_{[C ii]} + L_{[O i]} + L_{[Si ii]}), lowering the observed IR line-to-PAH ratio by Equation (2). However, this spatial decoupling of PAHs and ions is inconsistent with observations of PDRs in the Milky Way (Okada et al. 2013; Salgado et al. 2016), the Large Magellanic Cloud (Lebouteiller et al. 2012; Chevance et al. 2016), and local galaxies (Croxall et al. 2012; Abdullah et al. 2017; Bigiel et al. 2020). (4) Metallicity may influence the strength of cooling lines and the PAHs (Cormier et al. 2015; Smith et al. 2017; Croxall et al. 2017; Cormier et al. 2019; Aniano et al. 2020). However, direct effects on the PAH grains appear most pronounced at metallicities well below those seen in GOALS galaxies, which typically have Z/Z_⊙ ∼ 1–2 (Inami et al. 2013; Díaz-Santos et al. 2017). Therefore, the trends observed in Figures 4 and 5 likely reflect variations in the heating and cooling mechanisms on the scale of individual PDRs.

The photoelectric efficiency is not only a function of G/n_H, but also the size and ionization state of PAH grains (Bakes & Tielens 1994; Galliano et al. 2008; Tielens 2008). Observations of PDRs in the Milky Way indicate that _PE falls as the grain charging parameter (γ ≡ G₀ T^1/2/n_e ) rises, and grains become more ionized (Tielens 2008; Okada et al. 2013; Salgado et al. 2016). Indeed, the models of Bakes & Tielens (1994) shown in Figure 5 predict low _PE at high G/n_H, and the data fall closer to the model with a grain size distribution weighted more toward larger grains. Thus, a link between PAH properties and _PE is to be expected if the decrease in _PE at high Σ_IR arises from a change in the local physics of gas heating in PDRs as mediated by the charge and size of PAHs.

Ratios between the 6.2, 7.7, 8.6, and 11.3 μm PAH lines are well-established as tracers of grain size and ionization (e.g., Draine & Li 2001; Tielens 2008), and local LIRGs exhibit comparable ratios to nearby, normal star-forming galaxies and high-z ULIRGs (Stierwalt et al. 2014; Smith et al. 2007; Pope et al. 2008; Wu et al. 2010; Shipley et al. 2013). For example, star-forming LIRGs cluster tightly in L_{6.2 μm}/L_{7.7 μm} versus L_{11.3 μm}/L_{7.7 μm}, tracing grain size and ionization respectively, which show a larger scatter in sources with AGNs (Stierwalt et al. 2014).

Figure 6 shows _PE as a function of the L_{11.3 μm}/L_{7.7 μm} ratio, a tracer of grain charge, and the L_{7.7 μm}/L_{6.2 μm} ratio, a tracer of the average cationic grain size (Draine & Li 2001). We do not detect a correlation between _PE and the grain ionization state at a significant confidence level (Table 2); however, _PE and L_{7.7 μm}/L_{6.2 μm} are anti-correlated at a ∼4σ confidence level, albeit with large scatter, suggesting that that the average size of PAH grains may influence the observed photoelectric efficiencies in local (U)LIRGs, consistent with the sensitivity of theoretical photoelectric efficiencies on the grain size distribution (Figure 5).

Zoom In Zoom Out Reset image size

Smaller PAH grains emit more strongly in the shorter-wavelength bands, whereas larger grains are brighter at longer wavelengths (Allamandola et al. 1989; Schutte et al. 1993; Draine & Li 2001). Thus, the diagnostic utility of a PAH line ratio as a grain-size indicator is a function of the separation in wavelength of the two features. The 7.7/6.2 μm ratio has been a widespread tool in the literature as both features are bright in extragalactic sources, relatively unaffected by the 9.7 μm silicate ice feature, and were observable simultaneously with the IRS (Draine & Li 2001; O'Dowd et al. 2009; Sandstrom et al. 2012; Stierwalt et al. 2014). With the advent of AKARI/IRC, the 3.3 μm PAH feature became accessible for large numbers of extragalactic sources while unlocking a longer baseline diagnostic of grain size.

The ratio of 11.3 μm to 3.3 μm PAH intensity is one of the most robust tracers of PAH grain size because of the large baseline in wavelength (Allamandola et al. 1989; Jourdain de Muizon et al. 1990; Schutte et al. 1993; Mori et al. 2012; Ricca et al. 2012; Croiset et al. 2016; Maragkoudakis et al. 2020). Using the available AKARI spectra in GOALS, we plot _PE versus L_{11.3 μm}/L_{3.3 μm} in Figure 7 to further test if low _PE is associated with larger or smaller PAHs. We find that galaxies in the AKARI sub-sample of GOALS with low _PE show systematically higher values of 11.3/3.3 μm PAH emission (Table 2). This trend is not driven by systematic variations in the 11.3 μm feature strength, as the L_{11.3 μm}/L_PAH ratio is constant over the range in L_{11.3 μm}/L_{3.3 μm}, and exhibits minimal 1σ scatter of 0.05 dex about the average (Figure 7, bottom panel). Instead, the strength of the 3.3 μm features relative to L_PAH decreases dramatically by nearly an order of magnitude (Figure 7, center panel). The anti-correlation between _PE and L_{11.3 μm}/L_{3.3 μm} reinforces the importance of the mean grain size in dictating the observed photoelectric efficiency.

Zoom In Zoom Out Reset image size

The photoelectric heating efficiency of PAHs falls as the number of carbon atoms per particle increases (e.g., Bakes & Tielens 1994), and small grains are preferentially destroyed in the presence of harsh radiation fields (e.g., Jochims et al. 1994). This provides a simple physical interpretation of the data presented so far: PDRs in the most compact starbursts ($\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ ≳ 10.7) have, on average, more intense radiation fields per gas density (Díaz-Santos et al. 2017), which leads to a preferential destruction of the smallest PAH grains, raising the average grain size and leaving behind large PAHs capable of releasing both fewer and weaker photoelectrons. This decreases the energy of a typical photoelectron available to heat the gas, and results in a lower gas heating efficiency.

This scenario is consistent with principles of PDR evolution. The strength of the radiation field impinging upon PDRs (i.e., G) is inversely proportional to the distance between the ionization front and the ionizing photon source (Kaufman et al. 1999). Therefore, higher values of G/n_H are associated with younger PDR systems that have yet to evolve away from their host star, and are more likely to be found in galaxies with higher star formation rate surface densities (e.g., Díaz-Santos et al. 2017). The low photoelectric efficiencies at high Σ_IR are plausibly linked to higher fractions of young, short-lived, and dust-cocooned star-forming regions, where the high values of G/n_H destroy small dust grains, hindering the coupling between the radiation field strength and PDR gas temperatures. Such PDRs may be continuously replenished by the compaction of gas and dust during a major merger (Sanders et al. 1988; Hopkins et al. 2008). Indeed, the majority of late-stage mergers (80%) are above $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ ∼ 10.7 (Stierwalt et al. 2013; Díaz-Santos et al. 2017), where photoelectric efficiencies are low.

As shown in Figure 8, the L_{11.3 μm}/L_{7.7 μm} and L_{11.3 μm}/L_{3.3 μm} ratios in LIRGs and ULIRGs are consistent with predominantly ionized PAHs containing on average ∼60–150 C atoms per particle when compared to the modeled PAH spectra of Maragkoudakis et al. (2020). ¹¹ PAHs with N_C ≲ 40 tend to be photodestroyed in PDRs (Jochims et al. 1994), which is close to the lower bound on grain sizes we observe for LIRGs and ULIRGs in Figure 8. Other mechanisms for grain destruction include shock-induced fragmentation and/or shattering; however, interstellar shocks tend to destroy larger grains, which is the opposite effect we see in the data (Jones et al. 1996). Therefore, G/n_H may be a critical factor for determining the photoelectric efficiency in (U)LIRGs with the grain size distribution acting as an intermediary between the radiation field, and gas cooling and heating.

Zoom In Zoom Out Reset image size

The rise and fall of the cosmic star formation rate density may be accompanied by an increase in the efficiency of star formation at earlier times, although this remains the subject of debate (e.g., Lilly et al. 1996; Madau et al. 1996; Tacconi et al. 2010, 2013, 2018; Madau & Dickinson 2014; Genzel et al. 2015; Scoville et al. 2017b; Liu et al. 2019; Decarli et al. 2019). Nevertheless, starbursts above the main sequence have higher star formation efficiencies (star formation rate per unit molecular gas mass) than normal galaxies, independent of redshift (Saintonge et al. 2013; Genzel et al. 2015; Scoville et al. 2017b). Can changes in the global star formation efficiency of a galaxy, with redshift and/or distance from the main-sequence, be linked to variations in the ISM conditions on small scales?

Given that photoelectric heating is the dominant coupling mechanism between gas temperatures and stellar radiation fields, _PE could play a role in mediating the star formation efficiency today and in the past by raising or lowering the energy transfer from stellar photons into the gas. When the photoelectric efficiency is low, photoelectrons convert a lower fraction of the incident radiation field into raising the gas temperature. In other words, gas may remain cold despite overall less cooling relative to photoelectric heating because the mechanism by which gas heats in the first place is weak at low _PE. This may be more common at earlier times, because normal star-forming galaxies at high redshift are more compact than local galaxies at a given stellar mass (Buitrago et al. 2008; Conselice 2014; van der Wel et al. 2014; Mowla et al. 2019), and Σ_IR anti-correlates with _PE. In addition, efficient star formation in the thick disks of high-z dusty, star-forming galaxies could contain a number of star-forming regions that each resemble the central regions of local (U)LIRGs (e.g., Tacconi et al. 2008; Bothwell et al. 2010; Stacey et al. 2010; Brisbin et al. 2015), in which case high G/n_H and low _PE could be common in PDRs across a high-z starburst. These conditions may play a more important role at high z as the contribution of ULIRGs to the total star formation rate density increases from z = 0 to z = 2 by a factor of ∼21 (Murphy et al. 2011).

Recent efforts to combine mid- and far-IR measurements of gas heating and cooling at z ∼ 1–2 are largely limited to handfuls of ULIRGs with archival Spitzer/IRS spectra (Brisbin et al. 2015; McKinney et al. 2020). Using ALMA, McKinney et al. (2020) compared the [C ii] and 6.2 μm PAH emission for GS IRS20, a luminous, compact starburst galaxy at z = 1.9239 and found a low [C ii]/6.2 μm ratio at high IR surface density which could indicate a low photoelectric efficiency. This galaxy has a L_{[C ii]}/L_11.3 ratio of 0.11, which would correspond to a 11.3/3.3 μm ratio of 21.5 if z ∼ 2 galaxies follow similar scaling relations between the PAH features and [C ii] as in local (U)LIRGs. If this is the case, then GS IRS20 would have a 3.3 μm PAH flux of 7 × 10⁻¹⁷ W m⁻², which can be easily detected in under 15 minutes by the James Webb Space Telescope (JWST) Mid-Infrared Instrument (MIRI). While MIRI will be excellent at measuring bright PAH lines at high z, our ability to probe much of the rest-frame mid- and far-IR regime at high z remains limited. Future facilities, such as the Origins Space Telescope, ¹² a proposed flagship mission covering the near–far-IR with powerful imaging and spectroscopic capabilities and a large, cold telescope, would be a powerful tool for measuring the full energy budget in distant star-forming galaxies, and testing the idea that the star formation efficiency is linked to the balance of gas cooling and heating.