Excitation of Polycyclic Aromatic Hydrocarbon Emission: Dependence on Size Distribution, Ionization, and Starlight Spectrum and Intensity

We consider starlight from various stellar populations. We include unreddened spectra from the single-age stellar population ("starburst") models of Bruzual & Charlot (2003, hereafter BC03) for ages t ranging from 3 Myr to 1 Gyr. The BC03 models assume the stars to form with a standard initial mass function from gas with heavy-element mass fraction Z = 0.02 (i.e., near-solar metallicity). We also include one very low-metallicity model with Z = 0.0004 ≈ 0.02 Z_⊙ and t = 10 Myr to examine the effect of varying Z.

We also consider the "BPASS" single-age stellar population models (Eldridge et al. 2017; Stanway & Eldridge 2018), which include the effects of binary stars. We consider the same range of ages as for the BC03 models, and include one BPASS low-metallicity example (for t = 10 Myr, with Z = 0.001 ≈ 0.05 Z_⊙).

In addition to the BC03 and BPASS models, we consider the solar neighborhood spectrum as representative of the typical interstellar radiation field in the diffuse interstellar medium (ISM) of a star-forming galaxy with more-or-less steady star formation for the past ∼10 Gyr, with the starlight reddened by distributed interstellar dust. We use the model of Mathis et al. (1983; hereafter MMP) for the starlight in the solar neighborhood, but with slightly modified parameters (see discussion in Draine 2011b): the dilution factor for the 3000 K component is increased from W = 4 × 10⁻¹³ to 7 × 10⁻¹³, and the dilution factor for the 4000 K component is increased from W = 1.0 × 10⁻¹³ to 1.65 × 10⁻¹³. We refer to this as the modified MMP (mMMP) starlight spectrum, with energy density per unit frequency u_mMMP,ν , and total starlight energy density

$$\begin{eqnarray}&&{u}_{\mathrm{mMMP}}=1.043\times {10}^{-12}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 1 }$$

We also consider starlight from a very old stellar population, using the spectrum of stars in the bulge population of M31 adopted by Groves et al. (2012). The M31 bulge spectrum may be representative of the starlight heating dust in an elliptical galaxy.

Because we are considering the heating of dust grains in regions where the hydrogen is predominantly H i or H₂, the starlight spectra in all cases are cut off at the Lyman limit, h ν = I_H = 13.6 eV.

The starlight intensity will be characterized by the heating effect on the grains that dominate the far-infrared (FIR) emission. We calculate the rate of energy absorption by a specified "standard grain." For the standard grain, we adopt a 1.6:1 oblate "astrodust" grain (Draine & Hensley 2021), with porosity ${ \mathcal P }=0.2$ and effective radius a_eff = 0.1 μm. Let ${C}_{\mathrm{abs}}^{(\mathrm{Ad})}(\nu )$ be the orientation-averaged absorption cross section for this standard grain, c the speed of light, and u_⋆,ν the starlight energy density per unit frequency. The standard mMMP radiation field produces a heating rate

$$\begin{eqnarray}&&{h}_{\mathrm{ref}}\equiv \int d\nu \,{u}_{\mathrm{mMMP},\nu }\,c\,{C}_{\mathrm{abs}}^{(\mathrm{Ad})}(\nu )=1.958\times {10}^{-12}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 2 }$$

for our standard astrodust grain (a_eff = 0.1 μm, b/a = 1.6, ${ \mathcal P }=0.2$).

For each spectral shape u_⋆,ν we define a dimensionless parameter

$$\begin{eqnarray}&&{\gamma }_{\star }\equiv \displaystyle \frac{\left[\int d\nu \,{u}_{\star ,\nu }\,{C}_{\mathrm{abs}}^{(\mathrm{Ad})}(\nu )]/[\int d\nu \,{u}_{\star ,\nu }\right]}{\left[\int d\nu \,{u}_{\mathrm{mMMP},\nu }\,{C}_{\mathrm{abs}}^{(\mathrm{Ad})}(\nu )]/[\int d\nu \,{u}_{\mathrm{mMMP},\nu }\right]}.\end{eqnarray} \tag{ 3 }$$

γ_⋆ is the spectrum-averaged absorption cross section for the standard grain relative to the spectrum-averaged absorption cross section for the mMMP starlight spectrum. γ_⋆ is a measure of how effective a given radiation spectrum is (relative to the mMMP spectrum) for heating "standard" dust. By definition, γ_⋆ = 1 for the mMMP spectrum; γ_⋆ > 1 for bluer starlight (more readily absorbed by dust), and γ_⋆ < 1 for redder starlight (less effective for dust heating).

For each spectral shape, we define a reference energy density

$$\begin{eqnarray}&&{u}_{\mathrm{ref},\star }\equiv \displaystyle \frac{{u}_{\mathrm{mMMP}}}{{\gamma }_{\star }}\end{eqnarray} \tag{ 4 }$$

that produces the standard amount of heating (h = h_ref) for our standard grain. The intensity of radiation heating the dust in a region can be estimated from the wavelength of the FIR emission peak, which depends on the temperature (and therefore the heating rate) of the ∼0.1 μm grains that dominate the FIR emission. We characterize the heating effect of a radiation field u_⋆,ν by a dimensionless intensity parameter

$$\begin{eqnarray}&&U\equiv \displaystyle \frac{\int d\nu \,{u}_{\star ,\nu }\,c\,{C}_{\mathrm{abs}}^{(\mathrm{Ad})}(\nu )}{{h}_{\mathrm{ref}}}=\displaystyle \frac{{u}_{\star }}{{u}_{\mathrm{ref},\star }}={\gamma }_{\star }\displaystyle \frac{{u}_{\star }}{{u}_{\mathrm{mMMP}}}\end{eqnarray} \tag{ 5 }$$

where u_⋆ is the energy density of the radiation. With the definition in Equation (5), radiation fields with different spectra but the same U will heat the standard grain to the same temperature, resulting in the same FIR emission spectrum.

The "hardness" of the starlight is indicated by the mean energy per absorbed photon for a grain X,

$$\begin{eqnarray}&&\langle h\nu {\rangle }_{\mathrm{abs}}^{({\rm{X}})}\equiv \displaystyle \frac{\int d\nu \,{u}_{\star ,\nu }\,h\nu \,{C}_{\mathrm{abs}}^{({\rm{X}})}(\nu )}{\int d\nu \,{u}_{\star ,\nu }\,{C}_{\mathrm{abs}}^{({\rm{X}})}(\nu )}\,.\end{eqnarray} \tag{ 6 }$$

Table 1 gives 〈h ν〉_abs for our standard astrodust grain, and also for a PAH cation. As expected, the harder radiation fields (e.g., the 3 Myr old starburst) have larger $\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{Ad})}$. Because the opacity of the PAH nanoparticles rises more rapidly in the UV, $\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{PAH})}$ can be significantly larger than $\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{Ad})}$ when the radiation field is dominated by starlight from cool stars, as for the M31 bulge.

Table 1. Selected Starlight Spectra

Stellar Population	Ref.	γ⋆	uref,⋆ a	$\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{Ad})}$ b	$\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{PAH})}$ c
			(10−13 erg cm−3)	(eV)	(eV)
Z = 0.02, t = 3 Myr	d	5.72	1.82	6.73	8.64
"	e	5.45	1.92	6.54	8.30
Z = 0.0004, t = 10 Myr	d	5.73	1.82	6.70	8.57
Z = 0.001, t = 10 Myr	e	5.11	2.04	6.35	8.42
Z = 0.02, t = 10 Myr	d	4.64	2.25	5.90	7.85
"	e	4.58	2.28	5.84	7.80
Z = 0.02, t = 100 Myr	d	2.94	3.54	4.34	5.75
"	e	2.55	4.09	3.94	5.69
Z = 0.02, t = 300 Myr	d	1.70	6.13	2.98	4.59
"	e	1.27	8.19	2.38	4.29
Z = 0.02, t = 1 Gyr	d	0.798	13.1	1.63	3.04
"	e	0.675	15.5	1.41	2.95
mMMP ISRF	f	1	10.4	1.93	4.54
M31 bulge	g	0.580	18.0	1.22	2.68

Notes.

^a Energy density corresponding to U = 1. ^b For a = 0.1 μm 1.6:1 oblate astrodust with porosity ${ \mathcal P }=0.2$ (Draine & Hensley 2021). ^c For N_C = 105 PAH⁺. ^d BC03 (Bruzual & Charlot 2003). ^e BPASS (Eldridge et al. 2017; Stanway & Eldridge 2018). ^f mMMP (Mathis et al. 1983; Draine 2011b). ^g Groves et al. (2012).

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The starlight spectra, all scaled to give the same U = 1 heating rate for our standard astrodust grain, are shown in Figure 1. Because the t = 10 Myr Z = 0.0004 and t = 3 Myr Z = 0.02 starburst spectra are very similar for h ν < 13.6 eV (see Figure 1(a)), results calculated below for the t = 3 Myr Z = 0.02 spectrum may be taken to apply to the t = 10 Myr Z = 0.0004 case.

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In addition to studying the heating for the above-described starlight spectra, we also consider the case of starlight incident on dust clouds, with the radiation field within the cloud attenuated and reddened by intervening dust. As a representative case, we consider dust clouds with extinction A_V ≈ 2 mag. For the starlight radiation fields of interest, such a cloud is sufficiently thick that, in the absence of scattering, the bulk of the incident starlight energy will be absorbed by dust in the cloud and reradiated in the infrared. Grains near the cloud surface will be exposed to the unreddened spectrum; grains deeper in the cloud will be heated by a weaker and redder radiation field, and thus will be cooler.

We neglect scattering, and assume unidirectional radiation incident normally on one cloud surface. We take the extinction to have the wavelength dependence A_λ /A_V adopted by Hensley & Draine (2021), based largely on studies by Schlafly et al. (2016) and Fitzpatrick et al. (2019) for the diffuse interstellar medium (see Figure 2(a)). The starlight energy density per unit wavelength at a point within the cloud is taken to be

$$\begin{eqnarray}&&{u}_{\lambda }(x)={u}_{\lambda }(0){10}^{-0.4({A}_{\lambda }/{A}_{V}){A}_{V}x/L},\end{eqnarray} \tag{ 7 }$$

where 0 ≤ x ≤ L is the distance from the slab surface, L is the thickness of the cloud, and A_V is the extinction at V through the cloud. Equation (7) treats scattering like absorption for estimating the attenuation of the radiation field. For normally incident radiation, this will overestimate the attenuation. On the other hand, if some or all of the incoming radiation is incident at appreciable angles relative to the normal, the attenuation law of Equation (7) will tend to underestimate the attenuation within the cloud; with these two errors tending to partially compensate, we use Equation (7) to estimate the starlight intensity within slabs with total extinction A_V = 2 mag.

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For optically thin dust, we find the temperature distribution function (dP/dT)_j,a and calculate the time-averaged power per unit wavelength p_λ radiated by one grain:

$$\begin{eqnarray}&&{p}_{\lambda }^{(j)}(a)=4\pi {C}_{\mathrm{abs}}^{(j)}(a,\lambda )\int {dT}{\left(\displaystyle \frac{{dP}}{{dT}}\right)}_{j,a}{B}_{\lambda }(T),\end{eqnarray} \tag{ 12 }$$

where B_λ (T) is the usual blackbody function. Figure 4 shows emission per C atom from ionized and neutral PAHs of various sizes, when illuminated by the mMMP radiation field, and Figure 5 shows the emission per grain if heated by starlight from the M31 bulge population.

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Absorption of one photon can heat small PAHs to high enough temperatures that they can efficiently radiate in the shortest wavelength emission features (e.g., 3.3 μm, 5.27 μm, 5.7 μm, 6.2 μm) but larger PAHs radiate primarily in longer-wavelength features (Schutte et al. 1993). This is evident in Figures 4 and 5.

The DL07 PAH opacity model for cations includes features at 1.26 μm and 1.905 μm as recommended by Mattioda et al. (2005a). According to our models, the smaller PAH cations do emit a modest fraction of their energy in these features. These emission features appear unlikely to be observable on a galactic scale in the presence of stellar continuum at these wavelengths, but might be observable in reflection nebulae with JWST.

To better compare the relative efficiency of PAHs of different sizes for conversion of absorbed starlight energy into PAH emission features, Figure 6 shows how the strengths of various emission peaks depend on the PAH size by plotting ${(\lambda {p}_{\lambda })}_{\mathrm{peak}}/{p}_{\mathrm{total}}$ versus N_C, where p_total = ∫d λ p_λ is the total time-averaged power radiated by the PAH. Figure 6 is similar to Figure 6 of Draine & Li (2007), differing only because of (1) differences in the treatment of the small "continuum" opacity (see Equations (9–(11)), (2) a small change in the band strengths adopted for features at 14.19 μm and the 17 μm complex (see Appendix A), and (3) replacement of the Mathis et al. (1983) spectrum used by DL07 by the mMMP spectrum used here. We see that only the smallest PAH sizes (N_C ≲ 100) radiate appreciably in the 3.3 μm feature.

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Schutte et al. (1993) found that PAHs with 10²–10⁵ C atoms could radiate in the IRAS 25 μm photometric band. Figure 6 shows that the 17 μm feature is efficiently radiated by large PAHs, with N_C ≳ 10³. The strong 7.7 μm feature is efficiently radiated by PAHs with N_C ≈ 10². This strong feature plays a large role in observational studies of PAHs; because PAHs with N_C > 10³ are relatively inefficient at radiating in this feature, DL07 defined the PAH abundance parameter q_PAH to be the ratio of the mass in PAHs with N_C < 10³ to the total mass in dust.

The time-averaged spectrum emitted by a PAH depends on the spectrum of the starlight responsible for heating the PAH. Figure 7 shows the time-averaged emission from PAH neutrals and cations with N_C = 105 C atoms illuminated by different radiation fields, all with heating parameter U = 1. For this particular PAH size, the ratio I(6.2 μm)/I(7.7 μm) is approximately the same for all of these radiation fields, but the ratio I(3.3 μm)/I(7.7 μm) varies dramatically between heating by a young starburst spectrum and heating by an old stellar population (the M31 bulge spectrum). Thus interpretation of PAH emission from galaxies depends on the spectrum of the starlight heating the dust.

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The emission spectrum from a PAH depends on the distribution of photon energies absorbed by the PAH, but to a considerable extent it depends on a single number: the mean absorbed photon energy $\langle h\nu { \rangle }_{{\rm{abs}}}^{({\rm{PAH}})}$ per vibrational degree of freedom. A PAH containing N atoms has 3N − 6 vibrational degrees of freedom. If the illuminating spectrum is varied, so that $\langle h\nu { \rangle }_{{\rm{abs}}}^{({\rm{PAH}})}$ varies, we anticipate that the shape of the emission spectrum will be approximately constant if $\langle h\nu { \rangle }_{{\rm{abs}}}^{\left({\rm{PAH}}\right)}\approx {\rm{const}}$. Ricca et al. (2012) illustrated this with I(3.3 μm)/I(11.2 μm) versus N_C for two different initial energies.

In Figure 8 we vary N_C from 37 to 105 to keep $\langle h\nu { \rangle }_{\mathrm{abs}}^{(\mathrm{PAH})}/{N}_{{\rm{C}}}\,\approx 0.08\,\mathrm{eV}$ for the six different starlight spectra. Figure 8 shows the emission from neutral and ionized PAHs. The shape of the emission spectrum (e.g., I(3.3 μm)/I(11.2 μm)) is similar for all of the cases shown, as the radiation field heating the PAHs is varied from the very red M31 bulge spectrum ($\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{PAH})}=2.7\,\mathrm{eV}$) to the hard UV spectrum of a 3 Myr old starburst ($\langle h\nu {\rangle }_{\mathrm{abs}}^{(\mathrm{PAH})}=8.6\,\mathrm{eV}$).

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The spectrum of the starlight heating the dust will depend on the history of star formation, but can also be affected by reddening if there is sufficient dust present so that the galaxy is not optically thin in the ultraviolet. As an example, we consider dust clouds illuminated by stars external to the cloud.

Because the starlight heating the dust is a function of depth x into the cloud (see Equation (7)), the temperature distribution function (dP/dT)_j,a,x is a function of x in addition to depending on composition j and grain size a. To obtain the emission spectrum, we must integrate over the cloud volume. As a representative example, we consider clouds that can be approximated as slabs with total visual extinction A_V = 2 mag—this is large enough that most of the optical-UV energy incident on the cloud will be absorbed by dust in the cloud.

We consider a slab with A_V = 2 mag illuminated by radiation incident normally on one surface, with the energy density varying as in Equation (7). The emission by the dust is assumed to be optically thin. The average power per unit wavelength per grain of composition j and size a is obtained by integrating over the cloud volume:

$$\begin{eqnarray}&&\langle {p}_{\lambda }^{(j)}(a)\rangle =4\pi {C}_{\mathrm{abs}}^{(j)}(a,\lambda ){\int }_{0}^{L}\displaystyle \frac{{dx}}{L}\int {dT}{\left(\displaystyle \frac{{dP}}{{dT}}\right)}_{j,a,x}{B}_{\lambda }(T).\end{eqnarray} \tag{ 13 }$$

To calculate the total dust emission, we require the size distribution and optical properties for each dust component. We assume the overall dust mass to be dominated by grains composed of a mixture of amorphous silicate, other metal oxides, and hydrocarbons—the hypothetical material termed "astrodust" by Draine & Hensley (2021) and B. S. Hensley & B. T. Draine (2021, in preparation). For astrodust we assume a porosity ${ \mathcal P }=0.20$, and the astrodust mass/H estimated by Hensley & Draine (2021) and Draine & Hensley (2021). The astrodust volume per H in the diffuse ISM in the solar neighborhood, ${V}_{{\rm{Ad}}}=5.34\times {10}^{-27}{{\rm{cm}}}^{3}/{\rm{H}}$ , is given in Table 2.

Table 2. PAH Size Distribution Parameters ^a

Parameter	Standard Value	"Small PAHs"	"Large PAHs"
a01(Å)	4.0	3.0	5.0
B1(10−7 H−1)	6.134	18.80	2.893
qPAH	0.0379	0.0391	0.0351

Note.

^a All cases have: ${a}_{\min ,\ \mathrm{PAH}}=4.0\times {10}^{-8}\,\mathrm{cm}$, ${a}_{\max ,\ \mathrm{PAH}}=1.0\times {10}^{-6}\,\mathrm{cm}$, σ = 0.40, V_PAH,1 = 3.0 × 10⁻²⁸ cm³ H⁻¹, V_PAH,2 = 0.7 × 10⁻²⁸ cm³ H⁻¹, a₀₂ = 30 Å, B₂ = 3.113 × 10⁻¹⁰ H⁻¹.

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A simple power-law size distribution captures the overall balance between small and large grains, but more complex size distributions are required to reproduce the observed interstellar extinction (e.g., Kim et al. 1994; Weingartner & Draine 2001) and infrared emission (e.g., Draine & Anderson 1985; Desert et al. 1990; Li & Draine 2001; Siebenmorgen et al. 2014). Such size distributions are obtained for the "astrodust" model in a following paper (B. S. Hensley & B. T. Draine 2021, in preparation). However, to model the infrared emission from the astrodust component, here we assume a simple power-law size distribution

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{\mathrm{Ad}}}{{da}}=(4+p)\displaystyle \frac{(3/4\pi ){V}_{\mathrm{Ad}}}{{a}_{\max ,\mathrm{Ad}}^{4+p}-{a}_{\min ,\mathrm{Ad}}^{4+p}}\,{a}^{p},\\ \,{a}_{\min ,\mathrm{Ad}}\lt a\lt {a}_{\max ,\mathrm{Ad}}.\end{array}\end{eqnarray} \tag{ 14 }$$

We take p = −3.3, ${a}_{\min ,\mathrm{Ad}}=4.5\,\mathring{\rm A}$, and ${a}_{\max ,\mathrm{Ad}}=0.4\,\mu {\rm{m}}$ to approximately reproduce the average extinction law in the diffuse ISM. For each astrodust grain size, we calculate the probability distribution dP/dT for grain temperature. For the larger grains, dP/dT approaches a delta function.

DL07 modeled the PAH population as the sum of two log-normal size distributions:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{\mathrm{PAH}}}{{da}}=\displaystyle \sum _{j=1}^{2}\displaystyle \frac{{B}_{j}}{a}\exp \left\{-\displaystyle \frac{{\left[\mathrm{ln}(a/{a}_{0j})\right]}^{2}}{2{\sigma }^{2}}\right\},\\ \,{a}_{\min ,\mathrm{PAH}}\lt a\lt {a}_{\max ,\mathrm{PAH}}.\end{array}\end{eqnarray} \tag{ 15 }$$

About 75% of the PAH mass in the DL07 model was in a log-normal population with a₀₁ = 4.0 Å and σ = 0.40. These are the PAH sizes that are needed to account for the observed PAH emission features. We retain this population in the present work, with a volume V_PAH,1 = 3.0 × 10⁻²⁸ cm³ H⁻¹, containing C/H = 30 ppm.

Following DL07, we add additional carbonaceous material in a second log-normal size distribution. For this component, we take a₀₂ = 30 Å, large enough to contribute minimal emission for λ < 20 μm. We take this component to contribute a volume V_PAH,2 = 0.7 × 10⁻²⁸ cm³H⁻¹, containing C/H = 7 ppm. The two log-normal components together reproduce the observed strength of the 2175 Å feature.

Below some critical size, transient heating events will lead to "evaporation" of atoms or groups of atoms, leading to destruction of the nanoparticle. Guhathakurta & Draine (1989) estimated that graphitic clusters need to have ${N}_{{\rm{C}}}\geqslant 23$ to survive in the ISM. We adopt a lower cutoff ${a}_{\min ,\mathrm{PAH}}\,=4.0\,\mathring{\rm A}$, corresponding to ${N}_{{\rm{C}},\min }=27$ carbon atoms.

The size distributions are shown in Figure 9(a). The parameters in Table 2 correspond to C/H = 37 ppm in the PAH population. This differs from the value of 60 ppm adopted by DL07 because of recent findings that the extinction per H in the general diffuse ISM is smaller than previously thought. ⁹ For the standard parameters in Table 2, the fraction of the total dust mass contributed by PAHs containing N_C < 10³ C atoms is q_PAH = 0.038.

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Figure 2(b) shows the extinction curve corresponding to the size distributions of Figure 9(a). The detailed wavelength dependence of extinction is not accurately reproduced, but the model extinction deviates from the observed extinction by at most ∼20% (near ∼1400 Å), and generally much less.

Interstellar PAHs will be present in a range of charge states, including PAH⁻ anions, neutral PAH⁰ molecules, PAH⁺ cations, and PAH⁺⁺ dications. The present modeling collapses these to just PAH neutrals and PAH ions.

Because PAH neutrals and ions have quite different emission spectra (compare Figures 4(a) and (b)), we must specify the PAH ionized fraction as well as the size distribution. The PAH ionization will be determined by a statistical balance between photoionizations and collisional charging by collisions with electrons and ions, and will therefore depend on electron density and electron kinetic temperature in addition to the radiation intensity. Li & Draine (2001) estimated the ionized fraction f_ion(a), as a function of PAH radius a, for three sets of physical conditions for the diffuse interstellar medium: "cold neutral medium," "warm neutral medium," and "warm ionized medium." A weighted average of these was used for the overall ISM, and this ionized fraction was adopted by DL07.

To explore the sensitivity of overall emission spectra to assumptions regarding the ionization, we will consider three examples of f_ion(a), shown in Figure 9(b). Our "standard" case is similar to f_ion(a) assumed by DL07. The high and low f_ion(a) curves in Figure 9(b) correspond to factor-of-two shifts in the PAH radius for which f_ion = 0.5. Because the photoionization rate for a neutral PAH scales approximately as Ua³, and the electron capture rate for a PAH cation scales approximately as ${a}^{2}{n}_{e}/\sqrt{{T}_{e}}$, a shift of a factor of two in the size where f_ion = 0.5 corresponds approximately to a factor-of-two shift in the ratio ${Ua}\sqrt{{T}_{e}}/{n}_{e}$. It would not be surprising if even larger variations in PAH ionization occurred. The relative constancy of PAH spectra may therefore be an indication of self-regulation in the ISM that limits variations in $U\sqrt{{T}_{e}}/{n}_{e}$.

Let 4π j_λ be the power radiated per unit volume, per unit wavelength. The emissivity per H nucleus is obtained by summing the radiated power per grain over the grain size distribution:

$$\begin{eqnarray}&&\displaystyle \frac{4\pi {j}_{\lambda }}{{n}_{{\rm{H}}}}=\sum _{j}\int {da}\left[\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{j}}{{da}}\right]{p}_{\lambda }^{(j)}(a).\end{eqnarray} \tag{ 16 }$$

Figure 10 shows emission spectra calculated for the standard dust mixture heated by starlight with the mMMP spectrum, but varying U. At FIR wavelengths, the opacity of astrodust varies approximately as λ^−1.8 (Hensley & Draine 2021). Therefore, as U is increased, the temperature of the large grains increases as T ∝ U^1/5.8, with the wavelength where λ p_λ peaks in the FIR varying as

$$\begin{eqnarray}&&{\lambda }_{\mathrm{FIR}\,\mathrm{peak}}\approx 140\,\mu {\rm{m}}\times {U}^{-1/5.8}\,\,\mathrm{for}\,U\lesssim {10}^{4}.\end{eqnarray} \tag{ 17 }$$

At the same time, for U ≲ 10³, the shape of the emission spectrum at λ ≲ 15 μm remains nearly invariant, with the emission remaining dominated by cooling following single-photon heating. Only as U exceeds ∼10³ does the heating become sufficient to modify the shape of the λ < 15 μm emission, but the spectrum for λ < 10 μm remains unaffected even for U = 10⁴.

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For U ≳ 10⁵, the silicate features begin to appear in the emission spectrum (see Figure 10(b)), as the astrodust grains become warm enough (T ≳ 130 K) to radiate strongly in the 18 μm feature and, for U ≳ 10⁶, in the 9.7 μm feature as well. For U ≳ 10⁶, the silicate 10 μm emission feature is comparable to or stronger than the strongest PAH emission features.

The normalized emission $\lambda {j}_{\lambda }/{\left[\int {j}_{\lambda }d\lambda \right]}_{\mathrm{TIR}}=\lambda {L}_{\lambda }/{L}_{\mathrm{TIR}}$, where L_TIR is the total infrared power, is shown for the standard dust mixture in Figure 11 for three different starlight spectra. Results for U = 1 are shown in Figure 11(a), and for U = 10³ in Figure 11(b).

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The mMMP and M31 bulge spectra both have λ_{FIR peak} ≈ 140 μm for U = 1, but the 3 Myr old starburst spectrum has λ_{FIR peak} ≈ 130 μm, showing that the single parameter U does not completely capture the heating effects of the radiation field on a dust mixture—the shape of the spectrum also matters. Recall that U was defined in terms of the rate of heating of a_eff = 0.1 μm astrodust grains. For the standard dust mixture, the FIR emission from a_eff ≈ 0.1 μm dust appears to be representative of the broad FIR peak when the dust is heated by starlight spectra that are broadly similar to the mMMP spectrum. However, a very young starburst has a starlight spectrum with far more energy in the far-UV (see Figure 1), which has the effect of increasing the fraction of the starlight absorption contributed by the smaller grains. The smaller grains are somewhat warmer, hence the peak of the SED is at ∼130 μm for U = 1 rather than ∼140 μm as for the mMMP and M31 bulge spectra. Increasing U to U = 10³ shifts λ_{FIR peak} to ∼42 μm for the mMMP spectrum.

The PAH emission features are sensitive to the starlight spectrum. For our standard dust mixture, Figure 11 shows that the 3 Myr old starburst spectrum produces normalized PAH features that are ∼20% stronger than for heating by the mMMP spectrum, while heating by the M31 bulge spectrum produces normalized PAH features that are a factor of ∼2 weaker than for the mMMP spectrum, as previously found by Draine et al. (2014). The PAH band ratios (e.g., F(3.3 μm)/F(11.2 μm)) are sensitive to the spectrum of the illuminating starlight, and vary significantly as the radiation field is varied from a 3 Myr old starburst spectrum to the very red M31 bulge spectrum. The dependence of band ratios on the starlight spectrum is discussed below in Section 9.7.

Because the starlight heating the dust is a function of depth x into the cloud (see Equation (7)), the temperature distribution function (dP/dT)_j,a is a function of x, and we must therefore average over the cloud volume. The infrared emission is assumed to be optically thin. The cloud-averaged emissivity per H is

$$\begin{eqnarray}&&\left\langle \displaystyle \frac{4\pi {j}_{\lambda }}{{n}_{{\rm{H}}}}\right\rangle =\sum _{j}\int {da}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{j}}{{da}}\langle {p}_{\lambda }^{(j)}(a)\rangle .\end{eqnarray} \tag{ 18 }$$

with the cloud-averaged $\langle {p}_{\lambda }^{(j)}(a)\rangle$ given by Equation (13). Figure 12 shows normalized emission spectra for diffuse dust heated by starlight with U = 10³, and for a dust cloud with A_V = 2 exposed to the same radiation field at the cloud surface. Figure 12(a) is for starlight with the mMMP spectrum, appropriate for a mature star-forming galaxy. As expected, the FIR peak shifts from λ_{FIR peak} ≈ 40 μm for the diffuse ISM starlight to ∼50 μm for the cloud, because most of the starlight power absorbed in the clouds takes place in an attenuated (and reddened) radiation field, with dust temperatures somewhat lower than at the cloud surface. The 3.3 μm PAH feature drops by about a factor of two, because excitation of the 3.3 μm feature is dependent on the more energetic stellar photons, and these are attenuated relatively rapidly as one moves into the dust slab.

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Figure 12(b) shows the same calculation, but now for a radiation spectrum characteristic of a 3 Myr old starburst and U = 10³. Once again, attenuation within the cloud shifts the peak of the FIR emission longward by about a factor 1.2, but for this case the normalized PAH features are nearly unchanged. This is because most of the starlight energy is in the UV, and capable of exciting the PAH features via single-photon heating. Because the incident radiation field is dominated by FUV photons, the PAH excitation falls off like the overall dust heating rate, leaving the ratio of PAH emission to total emission almost unaffected. We will see below (in Section 9.2) that the PAH band ratios remain nearly unchanged by reddening when the unreddened starlight is dominated by UV.

Figure 13(a) shows how λ_{FIR peak} depends on the heating parameter U, and on the spectrum of the starlight. The wavelength λ_{FIR peak} depends primarily on U (see Equation (17)), but λ_{FIR peak} does depend slightly on the spectrum of the radiation heating the dust. Figure 13(b) shows λ_{FIR peak} for emission from an A_V = 2 mag cloud, as a function of the parameter U for the radiation at the cloud surface. The emission from the cloud includes radiation from dust grains heated by the attenuated radiation field within the cloud, which shifts λ_{FIR peak} longward of where it would be for only the emission from the grains at the cloud surface. Figure 13 shows that this shifts λ_{FIR peak} longward by about a factor 1.2 for the mMMP spectrum.

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Interpretation of the emission features in observed infrared spectra requires quantitative determination of the feature strengths. Feature "extraction" is complicated by the presence of a "continuum," and because the features may be overlapping in wavelength. In addition, the observed emission may be affected by extinction, particularly in the 9.7 μm silicate feature, which should be corrected for (Smith et al. 2007).

Various approaches have been taken. Some studies employ spline fits to the presumed continuum so that it can be subtracted to reveal the emission features (e.g., Peeters et al. 2002; Brandl et al. 2006). Another approach is to fit the observed spectra, including the continuum, using a physically motivated set of fitting functions with a tractable number of free parameters; PAHFIT (Smith et al. 2007) is one such spectral-fitting code. Different feature extraction techniques may differ appreciably in estimation of the power in the different PAH bands. Paper II (J.-D. T. Smith et al. 2021, in preparation) applies an expanded PAHFIT decomposition to our model spectra.

Component-fitting procedures such as PAHFIT provide the best measure of the power F radiated in a spectral feature, but require sophisticated fitting of a multicomponent model to the measured spectra. Since our modeled emission spectra do not include contributions from direct starlight, line emission, differential attenuation of the emission by dust (with the 10 μm silicate feature), and emission from other hot dust (e.g., from active galactic nuclei), a simpler approach can illustrate the main trends. Here we adopt a very simple method, to obtain the "clipped" flux F_clip in a feature: we specify points λ₁ and λ₂ on either side of the feature where the feature strength will be taken to be zero, define a "clip-line" $\lambda {F}_{\lambda }^{({\rm{c}}.{\rm{l}}.)}$ between these two wavelengths to be a linear function of $\mathrm{log}\lambda$ connecting λ F_λ at the clip points (see Figure 15), and define

$$\begin{eqnarray}&&{F}_{\mathrm{clip}}(\mathrm{band})\equiv {\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\left({F}_{\lambda }-{F}_{\lambda }^{({\rm{c}}.{\rm{l}}.)}\right)d\lambda .\end{eqnarray} \tag{ 19 }$$

Our adopted "clip points" λ₁, λ₂ for each feature or "band" are given in Table 3. We treat the broad 7.7 μm complex (with overlapping sub-features at 7.417, 7.598, and 7.85 μm) and the 8.6 μm feature as a single "7.7 μm" feature extending from 6.9 to 9.7 μm. Similarly, we aggregate emission features at 15.9, 16.4, 17.04, and 17.375 μm into a single "17 μm" feature extending from 15.5 to 18.5 μm. Values of F_clip(band)/F_TIR for our standard model heated by the mMMP ISRF with U = 1 are given for five features in Table 3.

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Figure 15(a) shows our model spectrum for dust heated by the mMMP starlight spectrum, with our standard PAH size distribution, and our standard PAH ionization fraction f_ion; Figure 15(b) shows the same dust model but heated by starlight with the spectrum of a 3 Myr old starburst and heating parameter U = 10³. We examine the strengths of the five features listed in Table 3. Figure 15 shows these five features with the adopted baselines. Because the real feature profiles may have broad wings (note the substantial power below the red "features" in Figure 15), the present approach will significantly underestimate the actual power in the features. The present approach does, however, provide a simple systematic way to quantify feature strengths, thereby allowing us to discuss variations of those feature strengths, and to compare to observed spectra.

The sensitivity of the different emission bands to the spectrum of the starlight heating the dust and PAHs is explored in Figure 16, which shows feature strengths (normalized by the total IR power) F_clip(band)/F_TIR for 24 different examples of starlight heating. All cases are for U = 1, but 12 different starlight spectra are considered. For each starlight spectrum, we compute the IR emission for diffuse dust, and also for dust in A_V = 2 mag clouds with the starlight incident on the cloud surface. For each of the single-age stellar populations (3, 10, 100, 300 Myr, and 1 Gyr), we consider both BC03 (Bruzual & Charlot 2003) and BPASS (Eldridge et al. 2017; Stanway & Eldridge 2018) stellar models. In Figure 16 the cases are ordered by decreasing fractional power in the 3.3 μm feature.

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As the (unreddened) radiation spectrum varies from a 3 Myr old starburst to the M31 bulge, F_clip(3.3)/F_TIR drops by a factor ∼6, from 0.91% to 0.15%. Other bands are less sensitive—F_clip(7.7)/F_TIR declines by only a factor ∼2.5, from 5.8% to 2.3%. Thus band ratios such as F_clip(3.3)/F_clip(7.7) are sensitive to the illuminating spectrum, as will be further examined below.

For the "cloud" cases, reddening of the starlight within the cloud results in the total emission from the cloud having fractional band powers F_clip/F_TIR that are lower than the values for the unreddened incident spectrum. For example, compare the mMMP ISRF and mMMP ISRF, A_V = 2 cases in Figure 16: F_clip(3.3 μm)/F_TIR is lower by a factor of ∼2 for the A_V = 2 cloud. However, if the incident starlight spectrum is dominated by UV, as for starbursts with ages ≲10 Myr, reddening leads to only a small decrease in F_clip(band)/F_TIR, because the starlight power is dominated by far-UV radiation—the PAHs and the dust are heated by the same photons. Even the UV-sensitive 3.3 μm band is minimally affected: F_clip(3.3 μm)/F_TIR is reduced by only 10% in going from unreddened BC03 3 Myr radiation to the case of an A_V = 2 cloud (see Figure 16).

The PAH abundance q_PAH can be estimated from the strength of the observed PAH emission features. q_PAH estimation is usually done using the 7.7 μm feature, because it is the strongest, and also because it was well-matched to band 4 of the Infrared Array Camera (Fazio et al. 2004), allowing the 7.7 μm feature to be measured efficiently by the Spitzer Space Telescope. q_PAH is taken to be proportional to the fraction of the total IR power appearing in the 7.7 μm feature:

$$\begin{eqnarray}&&{q}_{\mathrm{PAH}}={R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}\times \displaystyle \frac{{F}_{\mathrm{clip}}(7.7)}{{F}_{\mathrm{TIR}}},\end{eqnarray} \tag{ 20 }$$

where the factor ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ is obtained by modeling the PAH emission spectrum from a model with known q_PAH (e.g., Table 3). In this paper we extract feature fluxes F_clip using the simple method described in Section 9.1. The factor ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ will in general depend on the spectrum of the starlight responsible for heating the PAHs and dust, and also on both the state of ionization and the size distribution of the PAHs.

Figure 17(a) shows ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ for a number of different starlight spectra, and for three different assumed PAH ionization functions f_ion. While ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ varies among the different starlight spectra, it is gratifying to see that the variations in ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ are modest over a wide range of starlight spectra that might be appropriate in star-forming galaxies, ranging from a very young 3 Myr old starburst to a 300 Myr old starburst, with the mMMP radiation field falling in between: we can generally take ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}\approx {0.8}_{-0.25}^{+0.3}$ provided that the stellar population is not extremely evolved. For the very extreme case of the starlight from the M31 bulge population, we have ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}\approx 1.6;$ if this starlight is reprocessed by dust clouds with A_V ≈ 2, ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ rises to ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}\approx 2.8$.

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Figure 17(a) shows that variations in the PAH ionization also affect ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$, because PAH neutrals do not radiate as strongly as PAH cations in the 7.7 μm C–C band. Thus the low f_ion models have higher ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ values than the standard model. However, ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ is more sensitive to changes in the starlight spectrum than to variations in f_ion.

The sensitivity of ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ to the assumed size distribution is examined in Figure 17(b). ${R}_{{q}_{\mathrm{PAH}}}^{\mathrm{clip}}$ varies by only ∼±5% for a₀₁ = 4 ± 1 Å. From Figure 17 we see that if we have a good way to estimate the spectrum of the starlight exciting the PAH emission, we can estimate q_PAH to within ∼±10% accuracy from F_clip(7.7)/F_TIR.

As discussed in Section 8, the band strengths are also sensitive to the properties of the PAH population, particularly the size distribution and the fractional ionization f_ion(a). Because the peak temperature reached following the absorption of a single UV photon is determined by the heat capacity of the PAH, the emission spectrum depends on the PAH size. Figure 4 shows emission spectra for selected PAH sizes when illuminated by the mMMP ISRF: the emission shifts systematically to longer wavelengths as PAH size increases. In addition, the PAH properties may themselves change systematically with size. Shannon & Boersma (2019) discussed the effect of PAH size on the shape of the 7.7 μm complex.

The sensitivity to the PAH size distribution is explored in Figure 18(a), where filled symbols are for the standard size distribution, open squares are for the PAH size distribution shifted to peak at larger sizes, and open triangles are for the PAH size distribution shifted to smaller sizes. Note that in all cases we hold the lower cutoff fixed at ${a}_{\min }=4.0\,\mathring{\rm A}$.

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The 3.3 μm feature is sensitive to variations in the size distribution, because the 3.3 μm emission is dominated by the smallest PAHs—those with sufficiently small heat capacities such that a single ∼10 eV photon can heat the PAH to temperatures T ≳ 600 K where it can radiate effectively at 3.3 μm. Size distributions shifted to smaller sizes lead to relatively stronger 3.3 μm emission.

In the single-photon heating limit (U ≲ 10³), larger PAHs are most efficient for converting absorbed starlight energy into emission in the 17 μm feature (see Figure 6), and therefore shifting the PAH size distribution toward larger sizes raises the 17 μm feature strength relative to the other PAH features.

The sensitivity to the PAH ionization balance is explored in Figure 18(b); filled symbols are for the standard size-dependent ionization fraction f_ion(a), while open triangles and squares are for the "high" and "low" ionization fractions, respectively, illustrated in Figure 9. The model postulates that neutral PAHs have enhanced opacity (relative to PAH cations) in the 3.3 μm C–H stretch, and reduced opacity (relative to cations) in the 6.2 μm and 7.7 μm C–C stretching modes. Models with "low" f_ion therefore have stronger emission at 3.3 and 11.2 μm, while models with "high" f_ion have lower emission in those two bands. Models with high f_ion have increased emission in the 6.2 and 7.7 μm bands. Thus, band ratios such as F(3.3)/F(7.7) or F(11.2)/F(7.7) are diagnostic of the environmental conditions determining the PAH ionization balance.

At low starlight intensities, the PAH emission is excited by single-photon heating, and the fractional power emitted in each of the bands does not depend on the intensity of the starlight—only on its spectral shape. Figure 19 shows the fractional power F(band)/F_TIR in each of the five emission features as a function of U for our standard model, with Figure 19(a) calculated for the quite hard spectrum of a 3 Myr old starburst, and Figure 19(b) calculated for the mMMP spectrum. For the shortest wavelength bands (e.g., 3.3 μm), F_clip(band)/F_TIR remains relatively constant until U reaches very high values. For longer-wavelength bands, F_clip(band)/F_TIR begins to rise when some part of the PAH population remains warm enough between photon absorption events to be able to radiate in the band.

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As the starlight intensity parameter rises above U = 1, the first band affected is the 17 μm complex, with F_clip(17)/F_TIR initially rising, then dropping as U increases above ∼300, and then rising again for U > 10^4.5. The 17 μm feature is efficiently radiated by PAH nanoparticles with N_C ≈ 10³–10⁴ C atoms (see Figure 6), and for U ≳ 10² these nanoparticles do not fully cool between absorption events. Thermal emission at 17 μm requires temperatures such that h ν/kT ≲ 5, or T ≳ 170 K. Figure 20 shows energy distribution functions for PAHs with N_C ≈ 1200 and 5600, for selected values of U. For these two examples, for U = 10³ the nanoparticle spends ∼10% of the time above T > 150 K, able to radiate in the 17 μm feature. This accounts for the initial rise in F_clip(17)/F_TIR as U increases to ∼10². As U increases beyond ∼10^2.5, photons absorbed by an already-warm nanoparticle raise it to higher energies than it would have been able to reach by single-photon heating at lower U. Therefore, energy that for U ≲ 10² would be radiated in the 17 μm complex instead is shifted to shorter wavelengths, e.g., the 11.2 μm feature. This explains the drop in F_clip(17)/F_TIR as U is increased from 10² to 10⁴.

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The decrease in F_clip(17)/F_TIR to a minimum at U ≈ 10⁴ followed by a rise to a second peak at U ≈ 10⁶ is related to the bimodal size distribution adopted for the PAHs (see Figure 9(a)), with a second component (a₀₂ = 30 Å) having a mass distribution peaking near N_C ≈ 10⁵. The larger PAHs in this second component account for the second peak in F_clip(17)/F_TIR at U ≈ 10⁶. For very high U, the larger PAHs are heated to T ≳ 150 K (see Figure 20) and contribute to the 17 μm feature, accounting for the rise in F_clip(17)/F_TIR for U ≳ 10^4.5 in Figure 19. For smoother size distributions, the variation of F_clip(17)/F_TIR would have been reduced. Another complicating factor is that as U reaches ∼10³, the silicate material in the astrodust grains begins to radiate in the 18 μm silicate feature (see Figure 10(a)). The 18 μm silicate emission profile interferes with the simple method used here for extraction of the flux in the 17 μm feature, which assumes a simple "baseline" between 15.5 and 18.5 μm.

Similar behavior is seen for other bands. F_clip(11.2)/F_TIR initially rises, and then declines for U ≳ 10⁴ as the PAHs become hot enough to shift power to shorter wavelengths. F_clip(7.7)/F_TIR rises as U increases to ∼10^4.5, followed by a decline as power is shifted to shorter wavelengths. F_clip(6.2)/F_TIR and F_clip(3.3)/F_TIR have not yet peaked for the highest intensities U = 10⁶ considered here.

Above we have investigated how PAH band intensities, relative to total infrared (TIR), are affected by the spectrum and intensity of the starlight, and by the PAH size distribution and ionized fraction. Because the band intensities are proportional to the PAH abundance, which can vary, it is useful to see how PAH band ratios are affected by the starlight properties, and by the PAH size distribution and ionization. We emphasize that, although trends in ratios among features are conserved, the method employed to recover the feature strengths will affect the median ratio values, sometimes significantly. See Smith et al. (2021, in preparation), for a full suite of comparisons.

Lai et al. (2020) present 2.7–28 μm spectra of galaxies based on Spitzer and AKARI spectroscopy. Their "1C" sample consists of 60 galaxies drawn from the 113 galaxies in their "PAH bright" sample. The 1C sample galaxies were selected to have strong PAH emission but weak silicate features (either in absorption or emission) in order to minimize the effects of reddening. We have applied our simple feature extraction procedure to the Lai et al. (2020) 1C "template" spectrum after subtraction of emission lines from ions and H₂ (T. Lai 2020, private communication). The green diamonds in Figures 21(a)–(d) show the observed band ratios for the Lai et al. (2020) 1C galaxy sample. The error bars shown correspond to the first and ninth deciles for the 1C galaxy sample (T. Lai 2020, private communication).

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We have also applied the feature extraction method described in Section 9.1 to 25 galaxies from the Spitzer Infrared Nearby Galaxy Survey (SINGS; Kennicutt et al. 2003; Smith et al. 2007), after removing emission lines (see spectra in Appendix B). Attenuation by dust (though modest) was also corrected for. The band ratios for each galaxy are plotted as triangles in Figures 21(a)–(c). Some of the galaxy points are identified.

Each plot shows a grid for each of three starlight spectra: a 3 Myr old starburst, the mMMP local ISRF, and the M31 bulge stars. The 3 Myr old starburst and M31 bulge spectra span the range from very UV-bright to very red, with the mMMP spectrum falling in between (see Figure 1). The mMMP spectrum is a good estimate for a typical star-forming galaxy.

The model band ratios change when the illuminating stellar spectra are varied, with F_clip(6.2)/F_clip(7.7) dropping by a factor ∼1.5, F_clip(3.3)/F_clip(7.7) decreasing by a factor ∼2.5, and F_clip(17)/F_clip(7.7) increasing by a factor ∼2, as the starlight is varied from the 3 Myr starburst to the M31 bulge. Varying the PAH size distributions from "small" to "large" affects the band ratios similarly to changing the spectrum from the 3 Myr starburst to the M31 bulge.

The band ratios for the SINGS galaxies in Figures 21(a)–(c) are generally within the region spanned by the considered variations in PAH size, ionization, and starlight spectra. The "1C" galaxy sample has F_clip(6.2)/F_clip(7.7) in the middle of the model range, but F_clip(11.2)/F_clip(7.7) is lower than the models, and lower than 24 of the 25 SINGS galaxies shown. Similarly, the 1C galaxy sample has F_clip(17)/F_clip(7.7) lower than 24 of the 25 SINGS galaxies shown.

Because the 7.7 μm emission is primarily from PAH cations, F(11.2)/F(7.7) is expected to be sensitive to the PAH ionization, and this is seen in Figures 21(a) and (b). As expected, varying f_ion in the model has little effect on F_clip(6.2)/F_clip(7.7) (the 6.2 μm feature and the 7.7 μm complex are both attributed primarily to cations), but F_clip(11.2)/F_clip(7.7) decreases by a factor ∼1.5 as f_ion varies from the "low" to "high" examples in Figure 9(b). We see in Figures 21(a), (b) that our standard model heated by the mMMP starlight with U ≲ 10⁴ gives F_clip(11.2)/F_clip(7.7) and F_clip(6.2)/F_clip(7.7) close to observed values for the SINGS galaxies, and the considered variations in size distribution and f_ion appear able to accommodate the observed spread in F_clip(6.2)/F_clip(7.7) and F_clip(11.2)/F_clip(7.7).

Figure 21(c) shows F_clip(17)/F_clip(7.7) versus F_clip(6.2)/F_clip(7.7). F_clip(6.2)/F_clip(7.7) is relatively insensitive to ionization, because both features are thought to be dominated by cations. However, changing the size, or changing the starlight spectrum, does affect F_clip(6.2)/F_clip(7.7), as already seen in Figure 21(a).

Figure 21(d) shows how F_clip(3.3)/F_clip(7.7) and F_clip(3.3)/F_clip(11.2) respond to changes in illuminating spectrum, size distribution, and ionization. Harder spectra (e.g., the 3 Myr starburst) lead to higher values of F_clip(3.3)/F_clip(7.7) and F_clip(3.3)/F_clip(11.2) because the larger photon energies lead to higher peak temperatures, enhancing the 3.3 μm emission. Similarly, smaller grains lead to higher F_clip(3.3)/F_clip(7.7) and F_clip(3.3)/F_clip(11.2). Thus, F_clip(3.3)/F_clip(7.7) provides information on size and spectrum, while F_clip(11.2)/F_clip(7.7) (see Figures 21(a) and (b)) helps constrain f_ion. We note, however, that our model calculations tend to predict values of F_clip(3.3)/F_clip(7.7) that are significantly larger than those observed for the 1C sample—this is further discussed in Section 9.9 below. We also see in Figure 21(d) that the 3.3/11.2 band ratio is sensitive to the PAH size distribution, because only the smallest PAHs become hot enough to radiate at 3.3 μm (see Figure 6).

Emission features in the 16–19 μm range were reported in a number of Galactic objects by Van Kerckhoven et al. (2000). A characteristic 17 μm complex of features, first identified in NGC 7331 (Smith et al. 2004), is prominent in the emission from star-forming galaxies (Smith et al. 2007). While not yet definitively identified with specific PAH vibrational modes, its correlation with other PAH bands makes it appear likely that the 17 μm feature is also emission from PAHs. Van Kerckhoven et al. (2000) and Boersma et al. (2010) discuss the types of C–C–C bending modes that might be responsible for the 16–19 μm features. The DL07 PAH model used here includes opacity at 17 μm consistent with the observed 17 μm emission from galaxies.

Figure 21(b) shows F_clip(17)/F_clip(7.7) versus F_clip(11.2)/F_clip(7.7). The 1C sample average has F_clip(17)/F_clip(7.7) below the model grids, and lower than all but one of the SINGS galaxies shown. While most of the SINGS galaxies are consistent with our model grids, some have F_clip(17)/F_clip(7.7) below the model grids in Figure 21(c). The three galaxies with the lowest values of F_clip(17)/F_clip(7.7) are NGC 2798, NGC 3049, and NGC 3773. Figure 24 shows that each of these galaxies has a continuum that is strongly rising from 20 to 30 μm, indicative of heating by radiation fields with U ≳ 10³ (see Figure 10 for model SEDs calculated for U = 10³ and U = 10⁴). Figure 19 shows that F_clip(17)/F_TIR is expected to decrease as U is increased from 10³ to 10⁴, suggesting that high U values may explain the low F_clip(17)/F_clip(7.7) seen for some galaxies.

In Figures 22(a), (b) we show F_clip(17)/F_clip(7.7) and F_clip(11.2)/F_clip(7.7) calculated for U = 10³ and 10^3.5. If the heating rate parameter U is increased to 10^3.0 and 10^3.5, F_clip(17)/F_clip(7.7) is reduced, but F_clip(11.2)/F_clip(7.7) is hardly affected. This would be one way to lower F_clip(17)/F_clip(7.7) without changing the PAH properties or size distribution.

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Thus, the observed low values of F_clip(17)/F_clip(7.7) for NGC 2798, NGC 3049, and NGC 3779 might be explained by values of U consistent with the observed 20–30 μm continuum in these galaxies.

As already seen in Figure 21(a), the values of F_clip(11.2)/F_clip(7.7) and F_clip(6.2)/F_clip(7.7) observed for the SINGS galaxies are in approximate agreement with the model calculations. However, for our standard size distribution (a₀ = 4 Å), standard f_ion, and the mMMP starlight spectrum, the model calculations predict F_clip(3.3 μm)/F_clip(7.7 μm) ratios (see Figure 21(d)), or F_clip(3.3 μm)/F_clip(11.2 μm) ratios (see Figure 21(c)), that are significantly larger (by about a factor ∼2) than the observed values for the Lai et al. (2020) 1C spectra.

It is possible that the band strengths adopted for the 3.3 μm band are too large. However, the adopted band strengths (from DL07) ∫C_abs d λ⁻¹ = (3.94, 0.89) × 10⁻¹⁸ cm per C–H bond in (neutral, ionized) PAHs appear to be broadly consistent with results from theoretical calculations (see, e.g., Malloci et al. 2007; Bauschlicher et al. 2018; Yang et al. 2020). With our adopted band strength, the PAH nanoparticles in the model would account for only ∼1/3 of the observed interstellar 3.3 μm absorption feature. ¹¹ If the band strength adopted here is correct, and f_n ≈ 0.5, most of the observed 3.3 μm absorption must come from aromatic material in larger grains.

For the mMMP spectrum (which we suggest is appropriate for typical star-forming galaxies), the F_clip(3.3)/F_clip(7.7) ratio could be brought into agreement by shifting to the "large" size distribution (a₀ = 5 Å). However, we suspect that the overprediction of F_clip(3.3 μm)/F_clip(7.7 μm) may be mainly attributable to the neglect of other energy-loss channels: photoelectric emission, photodesorption, and fluorescence (Allamandola et al. 1989). Our calculation of the vibrational excitation of PAHs assumed that absorption of a photon converts the full photon energy h ν into vibrational excitation, with the vibrational energy then being removed only by infrared emission. However:

• 1.  
  High-energy photons can photoionize PAHs. When a photoionization takes place, only a fraction of the photon energy appears as "heat."
• 2.  
  The electronically excited state resulting from absorption of a UV photon may sometimes de-excite by luminescence: emission of an optical photon before "internal conversion" is able to transfer all of the electronic energy to the vibrational modes (see reviews by Witt & Vijh 2004; Witt & Lai 2020).
• 3.  
  A PAH with a high vibrational temperature may sometimes be able to radiate an optical photon via "Poincare fluorescence" (also known as "recurrent fluorescence"; Leger et al. 1988; Lai et al. 2017).
• 4.  
  A PAH with a large amount of vibrational energy per degree of freedom will sometimes break a C–H bond, ejecting a hydrogen atom (see, e.g., Marciniak et al. 2021). Neglect of the energy lost to bond-breaking will lead to overestimation of the 3.3 μm emission, which depends on the high-T tail of the temperature distribution function.

PAHs with our "standard" size distribution dn/da and ionization fraction f_ion(a), heated by the mMMP starlight spectrum thought to be appropriate for normal star-forming galaxies, may in fact be consistent with observations when the above energy-loss channels are included when calculating the PAH temperature distribution functions and emission. This will be the subject of future work.