© O. Berné et al. 2022

1 Introduction

In a seminal paper on the temperature of interstellar gas, it was proposed by Spitzer Jr (1948) that the photoelectric (PE) effect on dust grains provides an important source of heating of the neutral gas in galaxies. The central idea behind this PE heating mechanism is that the interaction with dust of ultraviolet (UV) photons from young stars, with energies up to the Lyman limit (E = 13.6 eV), releases electrons that can carry a couple eV of kinetic energy and thermalize with the gas, resulting in gas heating. Following the initial proposal of Spitzer, other authors proposed more detailed models of this mechanism (Jura 1976; de Jong 1977; Draine 1978). After the discovery that large molecules of the family of polycyclic aromatic hydrocarbons (PAHs) are ubiquitous in space (Leger & Puget 1984; Allamandola et al. 1985), it was proposed that these large molecules could contribute to the heating of the HI gas (D’Hendecourt & Léger 1987; Lepp & Dalgarno 1988). Verstraete et al. (1990) measured the photoionization yield of the PAH coronene, and estimated the contribution of a PAH population with a typical size of 80 carbons to the PE heating. They concluded that PE effect on PAHs represents a major contribution to the gas heating in cold diffuse clouds and a significant contribution in the warm phase. Bakes & Tielens (1994) and Weingartner & Draine (2001; respectively BT94 and WD01 hereafter) developed models that take into account grain size distributions from classical dust sizes (up to typically ~0.1 µm) down to the molecular (PAH) domain. BT94 concluded that about half of the gas heating is due to grains containing less than 1500 C atoms. While a proper description of PE heating in dust models is central to a number of astrophysical models and analysis, its implementation has only been benchmarked against observations in a limited number of studies. Deriving values of the PE heating efficiency for dust, e, from observations requires quantifying dust emission but also the total neutral gas cooling power from all relevant infrared (IR) cooling lines in particular from atomic oxygen and ionized carbon. Several studies have used observations from Herschel, SOFIA, and Spitzer to derive the cooling budget of the neutral gas (Bernard-Salas et al. 2015; Pabst et al. 2022; Joblin et al. 2018). Recent studies of Galactic photodissociation regions (PDRs) associated with star-forming regions have compared observationally derived e values with those calculated with the model of BT94 (Salgado et al. 2016; Salas et al. 2019; Pabst et al. 2022). There is a good qualitative agreement in terms of the shape of the curves but the calculated values are found to be larger than the observed ones by at least a factor of three. Since the earlier proposal that PAHs may provide a major contribution to the PE heating (D’Hendecourt & Léger 1987; Lepp & Dalgarno 1988; Verstraete et al. 1990), a number of observations have revealed a tight correlation between the [CII] flux and the PAH emission flux, which is in agreement with this proposal (Helou et al. 2001; Joblin et al. 2010; Lebouteiller et al. 2012). Joblin et al. (2010) mentioned that the PAH charge varies over the NGC 7023 nebula (Rapacioli et al. 2005; Berné et al. 2007) and this is expected to affect the contribution of PAHs to PE heating. Following this idea, Okada et al. (2013) provided the first study attempting to assess the connection between PAH charge and PE heating combining Spitzer and Herschel observations. The authors derived values of the ratio of gas emission to the sum of gas and PAH emission in several PDRs, and showed that its variation with the ionization parameter can be rationalized with a simple prescription based on previous modeling works (Tielens 2005; Weingartner & Draine 2001). This study concluded that PAHs provide a dominant contribution to the PE heating. However, it was conducted on unresolved sources which provided limited accuracy on the derived values and did not include any modeling of PAH ionization to support the conclusions. Modeling the charge state of PAHs and their contribution to the PE heating of the gas relies on the knowledge of the molecular properties of these species. Part of the relevant data have been gathered into models that describe the variation of the charge state of PAHs (e.g. Montillaud et al. 2013; Andrews et al. 2016). In addition, key molecular parameters that are needed to model the PE heating rate by PAHs of relevant sizes (i.e. >30 carbon atoms), both in neutral and cationic forms, are the photo-absorption cross sections [σ.sub.abs], the ionization yields Y, and the ionization potentials (IPs). The earlier values of Y obtained for neutral PAHs (Verstraete et al. 1990; Jochims et al. 1996) are now completed with values for cations (Wenzel et al. 2020). An additional fundamental parameter is the partition coefficient (D’Hendecourt & Léger 1987; Verstraete et al. 1990), here written 〈[γ.sub.e](E)〉, which determines, upon ionization by a photon of energy E, the fraction of the energy that is carried away in kinetic energy by the photoelectron relative to the available energy E − IP. In this regard, photoelectron spectral images upon UV ionization of coronene have been reported by Bréchignac et al. (2014).

In this article we present a study in which we evaluate in detail the importance of the ionization of PAHs as a source of neutral gas heating in galaxies. We first derive in Sect. 2 the observational diagnostics on the PAH charge state and the gas heating efficiency for PAHs. In Sect. 3, we present a simple analytical model of PAH ionization and gas heating by the PE effect on PAHs, which includes state-of-the-art data for molecular parameters as provided by laboratory measurements and quantum chemical calculations. In Sect. 4 we present the model results and comparisons to the models of WD01 and BT94. In Sect. 5, we compare the observational diagnostics with results obtained with our PAH model. In Sect. 6, we discuss the dominant role of PAH ionization as a source of gas heating, before concluding in Sect. 7.

2 Observational diagnostics of PAH charge state and efficiency of gas heating

In order to characterize the contribution of PE effect on PAHs to the gas heating, we derive two diagnostics from observations. The first diagnostic we are interested in is the ionization fraction of PAHs as derived from observations, R¡, which is determined by:

$$R_{\text{i}}=\frac{I_{{\text{PAH}}^{+}}}{I_{{\text{PAH}}^{+}}+I_{{\text{PAH}}^0}},$$(1)

where I_PAH+ and $I_{{\text{PAH}}^0}$ are the IR emission intensities attributed to cationic and neutral PAHs, respectively. In this study, we use the generic tool called PAHTAT (Pilleri et al. 2012) that allows one to derive $I_{{\text{PAH}}^{+}}$ and $I_{{\text{PAH}}^0}$ from spectroscopic IR observations¹.

The second diagnostic we derive, following Okada et al. (2013), is the ratio of gas emission to the sum of gas and PAH emission:

$$R_{\text{e}}=\frac{I_{\text{gas}}}{I_{\text{PAH}}+I_{\text{gas}}},$$(2)

with I_PAH the intensity of PAH emission and I_gas the intensity of neutral gas cooling lines (dominated by the far-infrared [CII] and [OI] lines, see Appendix A.3). When I_PAH ≫ I_gas, $R_{\text{e}}\approx \frac{I_{\text{gas}}}{I_{\text{PAH}}}$.

In addition to the two observational diagnostics described above, we derive the value of the ionization parameter:

$$\gamma =G_0\times \sqrt{T}/n_{\text{e}},$$(3)

where T is the gas temperature in K, and n_e is the electron density in cm⁻³. G₀ quantifies the intensity of the UV radiation field:

$$G_0=\frac{1}{Z}{\displaystyle {\int }_0^{\text{Ω}}{\displaystyle {\int }_{91.2\text{nm}}^{240\text{nm}}{I}_{\lambda }\left(\lambda \right)\text{d}\lambda \text{\hspace{0.17em}dΩ}},}$$(4)

where I_λ(λ) is the intensity of the radiation field in the Solar neighborhood as given by Habing (1968; here in W m⁻¹ sr⁻¹ nm⁻¹), Ω is the solid angle of this radiation field and Z = 1.68 × 10⁻⁶ W m⁻² is the normalization factor. This definition of G₀ is equivalent to that of Le Petit et al. (2006). The interval of integration over wavelengths of Eq. (4) corresponds to an energy range between 13.6 and 5.17 eV. The two observational diagnostics (R_i and R_e) and γ are derived for the NGC 7023 nebula in spatially resolved observations, and for a sample of Galactic and extragalatic sources described in the following sections.

2.1 The spatially resolved NGC 7023 NW PDR

NGC 7023 is a reflection nebula illuminated by the HD 200775 double system of spectral types B3V and B5 (Finkenzeller 1985; Racine 1968) situated at 320 pc from the Sun (Benisty et al. 2013). We are interested in the PDR which is located at 42″ north west (NW) of the star. NGC 7023 NW PDR is the most studied PDR and is seen almost edge-on. The dissociation front (i.e. the interface where H₂ is dissociated to 2 H atoms) is well seen (Fig. 1) through bright emission in the H2 (1-0) S (1) line at 2.12 µm, showing dense filamentary structures (n_H ~ 10^5–6 cm⁻³) embedded in a more diffuse gas (n_H ~ 10⁴ cm⁻³) (Joblin et al. 2018; Lemaire et al. 1996; Fuente et al. 1996; Chokshi et al. 1988). Several estimates point to a UV radiation field intensity at the dissociation front of G₀ = 2600 (Pilleri et al. 2012; Joblin et al. 2018; Chokshi et al. 1988). We derive the values of γ in NGC 7023 as a function of the distance to the star (see Appendix A.1) and summarize these values in Table 1. We extract the values of R_i and R_e (see Appendices A.2 and A.3, respectively), over the area delimited in Fig. 1.

The resulting maps are presented in Fig. 2. The derived ionization fraction for PAHs is found to decrease from regions near HD 200775 to regions beyond the dissociation front as the radiation field intensity decreases, as expected. This result is consistent with earlier studies (e.g., Pilleri et al. 2012; Boersma et al. 2016). R_e follows the opposite trend, that is it increases with distance to the star. This is in line with the trend observed by Okada et al. (2013) for three position in NGC 7023. Since PAH emission is directly related to the intensity of the radiation field, this variation of R_e is indicative of a better coupling between the radiation field and the gas heating in regions far from the star, where γ is low and PAHs are mainly neutral.

Fig. 1 General view of the NGC 7023 NW PDR. Grey image is Spìtzer-IRAC data at 8 µm. Green contours show the emission in the H2 1–0 S(1) line at 0.5, 1 and 2 × 10−7 W m−2 sr−1, from observations at CFHT presented in Lemaire et al. (1996). The orange contour defines the region considered here for the PE heating study.

2.2 Sample of Galactic and extragalactic sources

Values for R_e are derived from observations for a number of astrophysical environments for which we expect the PE effect on PAHs to be a major contributor to gas heating. These include star-forming regions, protoplanetary disks, star-forming galaxies, ultra luminous infrared galaxies, and a high-redshift galaxy. The methodology to derive these parameters is detailed in Appendix B. All derived values are summarized in Table 2 together with associated γ values, and are shown in Fig. 3. Similar to the trend observed in NGC 7023 NW, R_e values are found to be lower in sources with high γ values, such as protoplanetary disks or galaxies that actively form stars. However, the trend is also highly variable. For example, the Orion Bar and Ced 201 have similar γ values but their values of R_e differ by an order of magnitude.

3 Molecular model of the charge state and photoelectric emission from PAHs

Here we present a simple molecular model, whose objective is to compute the PAH ionization fraction, the heating efficiency for PAHs, and the heating rates by PAHs. This model is based on the photophysics of isolated interstellar PAHs and includes the latest results from laboratory experiments for PAH molecular parameters. Upon absorption of a UV photon, three main processes are expected to be in competition to relax the energy: ionization, dissociation, and radiative cooling (see, e.g., Joblin et al. 2020). Photoelectric heating is related to ionization, and therefore we focus on this process in the following.

Fig. 2 Maps of the ionization fraction of PAHs in NGC 7023 (upper panel), and ratio of gas cooling to PAH emission (lower panel).

Fig. 3 Measured emission ratio Re (Eq. (2)), as a function of physical conditions traced by the ionization parameter y, in a sample of astro-physical objects.

3.1 Selection of species and molecular parameters

A first step consists in selecting species for which molecular data is available. Here we rely on information given by the Theoretical spectral database of PAHs² (Malloci et al. 2007). The chosen molecules are listed in Table 3, having compact shapes and different characteristic sizes relevant for interstellar PAHs (Leger & Puget 1984; Leger et al. 1989; Allamandola et al. 1985; Montillaud et al. 2013; Andrews et al. 2016).

In the present model, we consider four charge states that is Z ∈ {−1,0,1,2}). Although there is no clear spectroscopic evidence for the presence of anions (Z ≤ −1) in the ISM, we include them because they could play a role in gas heating in environments with low γ values. Cations (Z = 1) and neutrals (Z = 0) have long been proposed to exist in PDRs (see review in Tielens 2008). Theoretical (Malloci et al. 2007) and experimental (Wenzel et al. 2020) data have shown that the dication stage (Z = 2) is accessible in PDRs for the considered species (Table 3), because their ionization potential IP² is lower than the Lyman limit of 13.6 eV. Data for IP³ values for PAHs are scarce, but Zhen et al. (2016) have shown that they are greater than 13.6eV for small PAHs, and Wenzel et al. (2020) demonstrate that this is probably the case up to a typical size of N_C ~ 50. Therefore, we do not consider trications in our model.

Values of the ionization potentials of a PAH with a given size N_C, are obtained using the empirical formalism of WD01, with updated parameters from Wenzel et al. (2020) as given in Appendix C.1. Several studies provide measurements of the ionization yields of neutral (Jochims et al. 1996; Verstraete et al. 1990) and cationic (Zhen et al. 2016; Wenzel et al. 2020) PAHs.

From these measurements, Jochims et al. (1996) and Wenzel et al. (2020) derived empirical laws for the ionization yields of neutral and cationic PAHs, respectively, which are used in this work (see Appendix C.2). The ionization yield curves are shown as a function of incident photon energy in Fig. 4 for three PAH sizes. Similarly to Visser et al. (2007), we use Y⁻(E) = 1. The threshold in energy for the detachment of the electron on anions is given by the electron affinity (e.g., Tschurl & Boesl 2006). The latter is very variable with the molecule but remains below 2 eV (see Theoretical spectral database of PAHs; Malloci et al. 2005; Wahab et al. 2022). We therefore assumed, as an approximation, that Y⁻(E) = 1 at all energies. We present another case in Appendix C.3.

Finally, we use the photoabsorption cross sections given by the Theoretical spectral database of PAHs³ (see also Malloci et al. 2004). First, we retrieve the photoabsorption cross section for each molecule in Table 3 and for each value of Z ∈ {−1, 0, 1, 2} and normalize it to the number of carbon atoms N_C. Second, for each value of Z ∈ {−1,0,1,2} we compute the averaged normalized photoabsorption cross section. This yields an average photoabsorption cross section per carbon atom σ(E, Z), from which the ionization cross section per carbon atom can be derived using:

$$\begin{array}{cc}{\sigma }_{\text{ion}}\left(E,Z\right)=Y^Z\left(E\right)\times \sigma \left(E,Z\right)& \left({\text{cm}}^2{\text{C}}^{-1}\right).\end{array}$$(5)

Figure 5 represents the σ(E, Z) and σ_ion(E, Z) curves as used in the model.

Fig. 4 Ionization yields YZ(E) for charge states Z = 0 (solid lines) and Z = 1 (dashed lines), computed from Eqs. (C.3)–(C.5) respectively, and for the three different PAH sizes. The yield for Z = −1, Y−(E) = 1 is not shown, for clarity of the figure.

3.2 PAH charge states and ionization fraction

At equilibrium the fractions f^Z of each PAH population in a Z charge state is given by:

$$\{\begin{array}{ll}\text{charge level\hspace{0.17em}Z}=-1:\hfill & f^{-}{k}_{\det }=f^0{k}_{\text{att}}{n}_e\hfill \\ \text{charge level\hspace{0.17em}Z}=0:\hfill & f^0{k}_{\text{pe}}^0+f^0{k}_{\text{att}}{n}_{\text{e}}=f^{+}{k}_{\text{rec}}^{+}{n}_e+f^{-}{k}_{\det }\hfill \\ \text{charge level\hspace{0.17em}Z}=1:\hfill & f^{+}{k}_{\text{rec}}^{+}{n}_e+f^{+}{k}_{\text{pe}}^{+}=f^{2+}{k}_{\text{rec}}^{2+}{n}_e+f^0{k}_{\text{pe}}^{+}\hfill \\ \text{charge level\hspace{0.17em}Z}=2:\hfill & f^2+k_{\text{rec}}^{2+}{n}_e=f^{+}{k}_{\text{pe}}^{+}\hfill \end{array}$$(6)

where $k_{\text{pe}}^Z$ and $k_{\text{rec}}^Z$ are respectively the ionization and recombination rates of PAHs in the Z ∈ {0,1,2} charge state. We note that when Z is an exponent in the notation, it is written {−, 0, +, 2+} rather than (−1,0,1,2}, to avoid confusion. We also introduce the attachment rate of an electron to a neutral PAH, $k_{\text{att}}\equiv k_{\text{rec}}^0$, and the detachment rate of the electron from an anion, $k_{\text{det}}\equiv k_{\text{pe}}^{-}$. In addition, the values of f^Z should fulfill the condition:

$${\displaystyle \sum _{{\text{Z}}_{\text{min}}}^{{\text{Z}}_{\text{max}}}{f}^{\text{Z}}=1.}$$(7)

For Z ∈ {−1,0,1,2}, Eqs. (6) and (7) yield

$$f^{-}={\left(1+\frac{k_{\det }}{k_{\text{att}}{n}_e}+\frac{k_{\det }{k}_{\text{pe}}^0}{k_{\text{att}}{k}_{\text{rec}}^{+}{n}_e^2}+\frac{k_{\det }{k}_{\text{pe}}^0{k}_{\text{pe}}^{+}}{k_{\text{att}}{k}_{\text{rec}}^{+}{k}_{\text{rec}}^{2+}{n}_e^3}\right)}^{-1},$$(8)

$$f^0={\left(1+\frac{k_{\text{att}}{n}_e}{k_{\det }}+\frac{k_{\text{pe}}^0}{k_{\text{rec}}^{+}{n}_e}+\frac{k_{\text{pe}}^0{k}_{\text{pe}}^{+}}{k_{\text{rec}}^{+}{k}_{\text{rec}}^{2+}{n}_e^2}\right)}^{-1},$$(9)

$$f^{+}={\left(1+\frac{k_{\text{rec}}^{+}{n}_e}{k_{\text{pe}}^0}+\frac{k_{\text{pe}}^{+}}{k_{\text{rec}}^{2+}}+\frac{k_{\text{att}}{k}_{\text{rec}}^{+}{n}_e^2}{k_{\det }{k}_{\text{pe}}^0}\right)}^{-1},$$(10)

$$f^{2+}={\left(1+\frac{k_{\text{rec}}^{+}{n}_e}{k_{\text{pe}}^{+}}+\frac{k_{\text{rec}}^{+}{k}_{\text{rec}}^{2+}{n}_e^2}{k_{\text{pe}}^0{k}_{\text{rec}}^{+}}+\frac{k_{\text{att}}{k}_{\text{rec}}^{+}{k}_{\text{rec}}^{2+}{n}_e^3}{k_{\det }{k}_{\text{pe}}^{+}{k}_{\text{pe}}^{+}}\right)}^{-1}.$$(11)

The ionization rates are given by:

$$k_{\text{pe}}^Z={\displaystyle {\int }_0^{\text{Ω}}{\displaystyle {\int }_{IP\left(Z+1\right)}^{13.6}\frac{{\sigma }_{\text{ion}}\left(E,Z\right)I_{\text{E}}\left(E\right)}{E}}\text{d}E\text{\hspace{0.17em}dΩ}}\left({\text{s}}^{-1}\right),$$(12)

where I_E(E), is the local intensity of the UV radiation field in W m² eV⁻¹ sr⁻¹.

The values of σ_ion(E, Z) have been defined above (Eq. (5), in particular). On the other hand, the recombination rate is calculated using Spitzer’s formalism (Spitzer 2004), adapted for PAH cations by Verstraete et al. (1990):

$$k_{\text{rec}}=1.28\times {10}^{-10}{N}_{\text{C}}\sqrt{T}\left(1+\text{Φ}\right)\left({\text{cm}}^3{\text{s}}^{-1}\right),$$(13)

where T is the gas temperature, and

$$\text{Φ}=\frac{eU}{k_{\text{B}}T}=\frac{1.85\times {10}^5}{T\sqrt{N_{\text{C}}}}\left(\text{dimensionless}\right).$$(14)

where e is the electron charge, U is the mean electrostatic potential evaluated at the radius of the PAH considered, defined by $U=\frac{1}{4\pi {\epsilon }_0}\frac{e}{0.9\times {10}^{-10}\sqrt{N_C}}$ (J C⁻¹), k_B the Boltzmann constant and T the gas temperature.

Equation (13) can be extended to include all charge states Z > 0 in the recombination rate:

$$k_{\text{rec}}^Z=1.28\times {10}^{-10}{N}_{\text{C}}\sqrt{T}\left(1+\text{Φ}\times \left(1+Z\right)\right)\left({\text{cm}}^3{\text{s}}^{-1}\right).$$(15)

For k_att, we use the work of Carelli et al. (2013) who provide an empirical fit to their quantum chemistry calculations that is

$$k_{\text{att}}=a{\left(\frac{T}{300\text{\hspace{0.17em}K}}\right)}^b\exp \left(-\frac{c}{T}\right)\left({\text{cm}}^3{\text{s}}^{-1}\right).$$(16)

For the parameters a, b, c we use the values determined by Carelli et al. (2013) for coronene, C₂₄H₁₂ which is the largest molecule in this study, that is a = 2.74 × 10⁻⁹ cm³ s⁻¹, b = 0.11, and c = −1.11. We note however that for all the molecules studied by Carelli et al. (2013), the attachment rate is of the order of ~10⁻⁹ cm³ s⁻¹ with only a small dependence on temperature. Hence this choice is unlikely to affect the model results.

Finally, we write the ensemble of charge fractions:

$$F=\left\{f^{++},f^{+},f^0,f^{-}\right\},$$(17)

and the modeled PAH positive ionization fraction is defined as:

$$f_{\text{i}}=f^{+}+f^{2+}.$$(18)

Fig. 5 Average photoabsorption (σ, solid line) and photoionization (σion, dashed line) cross sections per C atom of PAHs adopted in the 4 energy level model.

3.3 Photoelectic heating efficiency for PAHs

The power injected into the gas via the photoelectrons, P_e, is given by:

$$P_{\text{e}}=f^{-}\times P_{\text{e}}^{-}+f^0\times P_{\text{e}}^0+f^{+}\times P_{\text{e}}^{+}\left(\text{W}\right),$$(19)

where f⁻, f⁰ and f⁺ are the fractions of anionic, neutral and cationic PAHs, derived from Eqs. (8)-(10), respectively. $P_{\text{e}}^{-},P_{\text{e}}^0$ and $P_{\text{e}}^{+}$ are the powers injected in the gas by photoelectrons ejected from anionic, neutral and cationic PAHs, respectively, which are defined by:

$$\begin{array}{lll}{P}_{\text{e}}^Z\hfill & =\hfill & {\displaystyle {\int }_0^{\text{Ω}}{\displaystyle {\int }_{I{P}^{Z+1}}^{13.6}{\gamma }_{\text{e}}\left(E\right)\left(E-I{P}^{Z+1}\right)}}\hfill \\ \hfill & \hfill & \times {\sigma }_{\text{ion}}\left(E,Z\right)\frac{I_{\text{E}}\left(E\right)}{E}\text{d}E\text{\hspace{0.17em}\hspace{0.17em}dΩ\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\text{W}\right),\hfill \end{array}$$(20)

where γ_e(E) is the fraction of the energy that, after ionization, goes into kinetic energy of the photoelectron.

Energy conservation upon ionization requires:

$$E-IP=E^{*}\left(E\right)+E_{\text{K}}\left(E\right)+E_{\text{KM}}^{}\left(E\right),$$(21)

where E* is the internal energy of the PAH and is its kinetic energy, which, due to the much higher mass of the PAH compared to the electron and due to conservation of momentum, is negligible compared to the kinetic energy of the electron, EK. Next, following D’Hendecourt & Léger (1987), we define the partition coefficient 〈γ_e(E)〉 as the energy averaged γ_e(E):

$$\langle {\gamma }_{\text{e}}\left(E\right)\rangle =\langle \frac{E_{\text{K}}\left(E\right)}{E-IP}\rangle .$$(22)

We estimate the value of 〈γ_e(E)〉 from photoelectron spectroscopy measurements performed using VUV synchrotron radiation and for the coronene molecule (Bréchignac et al. 2014).

The derived value of 〈γ_e(E)〉 = 0.46 ± 0.06 (68% confidence interval) is found to be in agreement with the value of 〈γ_e(E)〉 = 0.5, which was used in a number of articles (e.g., Verstraete et al. 1990; Tielens 2005; Bakes & Tielens 1994) and derived from older measurements on benzene by Terenin & Vilessov (1964; see Appendix C.4). For anionic PAHs, we assume that all energy available is taken by the electron. For IP⁰ = 0, this means that all the energy of the photon is transformed into kinetic energy of the electron.

On the other hand, the power absorbed, mostly (but not only) in the UV, by each PAH charge state, $P_{\text{Rad}}^Z$, is given by:

$$P_{\text{Rad}}^Z={\displaystyle {\int }_0^{\Omega }{\displaystyle {\int }_0^{13.6\text{eV}}\sigma \left(E,Z\right)I_E\left(E\right)\text{d}E\text{dΩ\hspace{1em}}\left(\text{W}\right).}}$$(23)

The total radiative power absorbed by PAHs, P_Rad, is then defined by:

$$P_{\text{Rad}}=f^{-}\times P_{\text{Rad}}^{-}+f^0\times P_{\text{Rad}}^0+f^{+}\times P_{\text{Rad}}^{+}+f^{2+}\times P_{\text{Rad}}^{2+}\left(\text{W}\right).$$(24)

We define the heating efficiency for PAHs, ϵ_PAH, as the ratio between the power injected in the gas via photoelectrons from PAHs to the total power of the radiation absorbed by PAHs, that is

$${\epsilon }_{\text{PAH}}=\frac{P_{\text{e}}}{P_{\text{Rad}}}.$$(25)

We note that the derived values for the radiative power and heating efficiency depend on N_C, through absorption and ionization cross sections, and through recombination rates with electrons. For the sake of simplicity this variable has, however, not been included in the notation. Also, our definition of the PE heating efficiency for PAHs is similar to that used by WD01 to compute their heating efficiency for grains, ϵ_Γ, but differs from the definition of e by BT94, who considered only absorbed photons in the 6–13.6 eV range. Finally, the PE heating rate by PAHs can be derived using the following relation:

$${\text{Γ}}_{\text{PAH}}=P_e\times \frac{f_{\text{C}}\left[C\right]}{N_C}={\epsilon }_{\text{PAH}}{P}_{\text{Rad}}\frac{f_{\text{C}}\left[C\right]}{N_C}\left(\text{W\hspace{0.17em}}{\text{H}}^{-1}\right),$$(26)

where f_C is the fraction of cosmic carbon locked in PAHs, and [C] is the cosmic abundance of carbon relative to hydrogen. The python code to compute Γ_PAH and ϵ_PAH with the model described in this section is provided through the Cosmic PAH portal⁴ or directly online⁵.

4 Model results and comparison with previous models

4.1 Charge balance

Figure 6 shows the variation of the population fractions of PAHs in charge state Z, as a function of γ, derived from the model, for T = 1000 K, N_C = 54, and stellar effective temperature of T_eff = 3 × 10⁴ K using a Kurucz (1993) star model. For γ ≲ 10³ (e.g. diffuse ISM or dark clouds), the population is dominated by neutral PAHs, with an anionic contribution up to 20%. A comparable fraction of neutrals and anions was also derived at low γ values by both BT94 and WD01 for 5 Å grains (comparable to PAHs of size N_C = 54). Neutral PAHs dominate up to γ ~ 10⁴, followed by cations with a maximum at about γ ~ 3 × 10⁴. Finally, dications are the dominant species for γ ≳ 10⁵. For γ = 10⁵ and the same stellar effective temperature, WD01 find values for the charge fractions (Eq. (17)) of F = {0.7,0.2,0.1,0.0}. We infer comparable fractions but with less dications and more cations, F = {0.5,0.4,0.1,0.0} (Fig. 6). The difference results from the higher ratio of ionization rate to recombination rate in WD01 compared to our model for high γ values. On the contrary, with the same stellar effective temperature and the same gas temperature, BT94 find at γ = 4 × 10⁴ F = {0,0.4,0.6,0.0}, and F = {0.2,0.5,0.3,0.0}, for planar and spherical particles, respectively, resulting in lower ionized fractions compared to our model with F = {0.4,0.5,0.1,0} (Fig. 6).

Fig. 6 Fractions in the different PAH charge state Z =−1,0,1,2 as a function of the ionization parameter y, which were derived using our PAH model and Teff = 3 × 104 K.

Fig. 7 Photoelectric efficiency ϵPAH computed for NC = 54, Teff = 30 000 K, and T = 100 and 1000 K with the PAH heating model. For comparison, we give the photoelectric efficiency ϵΓ computed by WDOl for grains with a size a = 4.9 A equivalent to NC = 54 (Eq. (C.2)), and the heating efficiency for PAHs given in Tielens (2021).

4.2 Heating efficiency

In Fig. 7, we present the results of the computation of ϵ_PAH with the PE heating model by PAHs for T_eff = 30 000 K, N_C = 54, and two values of the gas temperature, T = 100 K and T = 1000 K. We compare ϵ_PAH with the efficiencies ϵ_Γ derived for the same physical conditions by WD01, for grains of size a = 4.9 Å, corresponding to N_C = 54 (Eq. (C.2)). For γ = 100 we find a good agreement between the two models, for both gas temperatures. For higher values of γ, ϵ_PAH is systematically higher than ϵ_Γ. The difference increases with increasing γ, up to about an order of magnitude for values of γ ≳ 10⁵. This can be rationalized by a slower decrease of the ionization to recombination rate ratio with γ in our model as compared to WD01 (see previous section). Tielens (2021) provides the following formula for the PAH heating efficiency:

$${\epsilon }_{\text{PAH}}=\frac{0.06}{1+7\times {10}^{-5}\gamma },$$(27)

which we also display in Fig. 7. This curve is also below the heating efficiency we infer for PAHs. At γ ≲ 10³ this is due to the absence of photo-detachment in Eq. (27). For larger values of y ≳ 10³, the difference stems, notably, from the use of a photoelectron yield Y(E) = 0.3 at 10 eV by Tielens (2021), while we use a value closer to Y(E) = 0.5 (see Fig. 4). Since BT94 do not provide the photoelectric heating efficiency curves for different sizes, we cannot directly compare our results to their model.

Fig. 8 Photoelectric heating rate of the gas. Dash-dotted line: average model from Weingartner & Draine (2001; see text for details). Dashed line: model from Bakes & Tielens (1994). Continuous lines: PAH model (this work) for two PAH abundances, that is fC = 0.05,0.1.

4.3 Heating rates

The PAH heating rate we derived as a function of γ is shown in Fig. 8 for T = 100 K, N_C = 54, T_eff = 3 × 10⁴K, and two PAH abundances. We compare these values to the total heating rate of WD01 averaged over several dust size distributions (Fig. 15 of WD01) and that of BT94, obtained for the same physical conditions. Overall, the curves agree well. This implies that, with a purely molecular model, PAHs can produce a heating rate comparable to that of earlier models including a full size distribution of dust grains. We can also compare the results of our model with those of Verstraete et al. (1990) for the diffuse ISM. The physical conditions are Go = 1.7 with the Draine (1978) field, a gas temperature of T = 80 K, and an electron density n_e = 3 × 10⁻² cm⁻³ yielding γ ~ 500 K^1/2 cm⁻³. With a value of f_C = 0.1, as in Verstraete et al. (1990), our model yields Γ_PAH = 2.7 × 10⁻³³W H⁻¹ (Table 4). The latter authors find a value that is larger by a factor ~3, that is Γ_PAH = 8 × 10⁻³³W H⁻¹. This is probably due to a combination of the following factors: (1) the higher ionization yield used by these authors, for example Y ~ 0.7 at 10 eV compared to Y ~ 0.5 in our model (Fig. 4), 2) their larger value for the partition coefficient 〈γ_e(E)〉 = 0.5 compared to 0.46 in our model, 3) the fact that we include the ionization of cations in our model, which affects the charge balance and reduces the heating rate, because the dications do not heat the gas.

Fig. 9 Observed emission ratio Re (Table 2) and modeled photoelectric heating efficiency for PAHs, ϵPAH, as a function of the ionization parameter γ, for various stellar effective temperatures. The upper curve corresponds to Teff = 4 × 104 K, and the lowest curve to Teff = 104 K. All models use NC = 54 and a gas temperature of T = 500 K.

4.4 Empirical formulas for the heating efficiency and rate

We computed ePAH using the molecular model presented in Sect. 3, for a fixed size N_C = 54 (corresponding to cicumcoronene, see Table 3), and a gas temperature T = 500 K, using radiation fields with stellar spectral types at various effective temperatures (T_eff in the 10⁴–4 × 10⁴ K range). For T_eff ≥ 3 × 10⁴, we used stellar spectra from the Pollux database⁶ (Palacios et al. 2010) computed with the CMFGEN code (Hillier & Miller 1998). The lower temperature stellar spectra are from Kurucz (1993) database. The resulting PE efficiency curves for PAHs as a function of the ionization parameter $\gamma =G_0\sqrt{T}/n_e$ are shown in Fig. 9. The values of ϵ_PAH can be fitted using the following empirical formula:

$${\epsilon }_{\text{PAH}}\approx \frac{a}{1+b\times \gamma }+\frac{c}{1+d\times \gamma }.$$(28)

This is similar to the expression, proposed by Tielens (2005; see Eq. (27)) but includes an additional term to take into account photo-detachment at low γ values. We fit the analytical expression of Eq. (28) to the model derived values ϵ_PAH (Fig. 9) for all radiation fields, and provide the obtained values of the parameters in Table 5. Using Eq. (24) we computed P_Rad for N_C = 54 for T = 500 K and T_eff between 10⁴ and 4 × 10⁴ K, as a function of G₀. The result of this calculation, combined to Eq. (26) using N_C = 54, and [C] = 2.7 × 10⁻⁴ per H atom (Tielens 2021) yields an average heating rate for PAHs:

$${\text{Γ}}_{\text{PAH}}=3.0\pm 0.6\times {10}^{-31}{\epsilon }_{\text{PAH}}{G}_0{f}_C\left({\text{WH}}^{-1}\right),$$(29)

where the uncertainty on the numerical coefficient is the standard deviation among all numerical values obtained with the model.

5 Charge state and photo-electric heating of PAHs: Model vs. observations

In this section we discuss the comparison between the observational results presented in Sect. 2 and the results of the molecular model described in Sect. 3. More specifically, for the case of NGC 7023, we compare R_i, the ionization fraction of PAHs derived from the analysis of the mid-infrared spectra of the PDR (see Appendix A.2) to f_i, the modeled PAH ionization fraction. We also compare the emission ratio, R_e, to computed values of ϵ_PAH. If PAHs are the main source of gas heating, then the later two values and their variations should be in close agreement. We then extend this comparison to our sample of galactic and extragalactic sources. In addition, for NGC 7023, the Orion Bar, and the diffuse ISM, we compare the total gas cooling Λ_gas to the modeled heating rate by PAHs Γ_PAH.

5.1 Comparison of model results with observations in NGC 7023

We first computed the model for three PAH sizes that is N_C = 96, 54, 32, using the physical parameters in Table 1. and the stellar spectrum of HD 200775 (see Appendix A.1). The gas temperature in NGC 7023 NW varies between 750 and 150 K. We therefore adopt a fixed value of T = 500 K, since this parameter has only a small effect on the model results (Fig. 7). The comparison of model results with the observational diagnostics is provided for the eight positions listed in Table 1. In Fig. 10 we compare the ionization fraction obtained with the model, f_i, to those derived from observations in NGC 7023, R_i, as a function of γ. The obtained good agreement suggests that the model is able to quantify the evolution of the charge state of PAHs in NGC 7023. The PAH size plays a small role in the values of f_i, but better results are obtained for PAHs with N_C ~ 54. In Fig. 11, we compare R_e as measured in NGC 7023 with values of the heating efficiency, ϵ_PAH, derived from the model for three PAH sizes. The agreement in terms of curve shape and absolute values is good, with all observed values intersecting the model, except for the closest positions to the star. There are a number of parameters that could explain this discrepancy including less precise values for the physical conditions in this region, or a possible additional source of heating in this environment, such as shocks due to the expansion of a small HII region. This HII region could also contribute to some fraction of the [CII] emission that we include in the total cooling. Overall, however, R_e compares well to ϵ_PAH∙ Finally, we can also compare the total photoelectric heating rate by PAHs Γ_PAH to the total gas cooling rate Λ_gas = 4π × I_gas/N_H, with N_H the total hydrogen column density. Joblin et al. (2010) provide values for N_H in the NGC 7023 PDR, which are N_H = 10.5,8.1 and 3.9 × 10²¹ cm⁻², at 50,48 and 47″ from the star, respectively. We thus adopt N_H = 7.5 ± 3.3 × 10²¹ cm⁻². At distances 45–50″ from the star, we derive I_gas = 3.8 ± 1.6 × 10⁻⁶ W m⁻² sr⁻¹. This yields a cooling rate Λ_gas = 6.4 + 4.6 × 10⁻³¹ W H⁻¹ for this range of distances. The heating rates by PAHs derived from the model for the physical conditions at 45 and 50″ (Table 1) and a value f_C = 0.05 are, for both positions, Γ_PAH = 6.5 × 10⁻³¹ W H⁻¹, in agreement with the cooling rates derived from the observations.

Fig. 10 Variation with γ of the PAH ionization fraction Ri in NGC 7023 (diamonds with errobars). Values derived from the map in Fig. 2 for the eight angular distances between 20 and 55″ from HD 200775 given in Table 1. The values of fi from the PAH model are shown with lines for a gas temperature T = 500 K, and three PAH sizes.

Fig. 11 Variation with y of the emission ratio Re in NGC 7023 derived from the map in Fig. 2 for the eight angular distances between 20 and 55″ from HD 200775 given in Table 1. The values of ϵPAH from the PAH model are shown with lines for a gas temperature T = 500 K, and three PAH sizes.

5.2 Comparison of model results with observations for the sample of Galactic and extragalactic regions

5.2.1 Photoelectric heating efficiency

Figure 9 compares the variation with γ of the observational diagnostic R_e to the modeled values of ϵ_PAH for the set of objects listed in Table 2. Here we adopt N_C = 54 in the model, and a gas temperature T = 500 K, but we note that the choice of values for these parameters has a limited effect on the model results for γ ≳ 10³ (see Figs. 10, 11, and 7), applicable to most sources. In contrast, since the effective temperature of the radiation field has strong effects on the value of ϵ_PAH (see Spaans et al. 1994), we compute the models with appropriate effective temperatures. Figure 9 shows that, in general, there is a good agreement between R_e and ϵ_PAH, in terms of curve shape but also of absolute values. The upper part of Fig. 9 presents the comparison between R_e and ϵ_PAH for interstellar regions irradiated by the most massive stars with 20 000 ≤ T_eff ≤ 40 000 K. In the Orion Veil, R_e is somewhat lower than the predicted ϵ_PAH, for a star with T_eff = 40 000 K (which is appropriate for Θ¹ Ononis C). This lower value can be explained by the fact that the radiation field impinging on the Veil is likely softer than that of Θ¹ Orionis C, because of the UV absorption by dust and PAHs situated in between the Veil and the massive star. The Orion Bar which is much closer to Θ¹ Orionis C has a value of R_e which is in good agreement with that of ϵ_PAH for T_eff = 40 000 K. The other nebulae illuminated by O or early B stars in the sample, that is the Horsehead and IC 63 also have values of R_e in good agreement with modeled values of ϵ_PAH for T_eff = 30 000–40 000 K. For the LMC, R_e is in good agreement with the ϵ_PAH curve for the most massive stars (T_eff > 3 × 10⁴ K). This is compatible with the fact that the values we use to derive R_e from Rubin et al. (2009) and Lebouteiller et al. (2012) concern star-forming regions. R_e in star-forming galaxies falls within the range of the models with T_eff ~ 25 000 K, implying an average radiation field which is softer than that of O stars and dominated by B-type stars. R_e for ULIRGs is also in good agreement with the model. ULIRGs have a lower heating efficiency as compared to star-forming galaxies of the local universe, which is well explained in the model by the higher value of the ionization parameter γ in these galaxies due to intense massive star formation, increasing the average integrated intensity of the radiation field G₀ (Díaz-Santos et al. 2017; McKinney et al. 2021). The R_e value of the high redshift galaxy GS IRS20 is also in good agreement with the model. Figure 9 also presents the comparison between R_e and ϵ_PAH for interstellar regions irradiated by intermediate-mass stars with 10000 ≤ T_eff ≤ 15 000 K. This include NGC 7023 which was presented in detail in Sect. 5.1. The case of Ced 201 is particularly interesting, because it has a low PAH heating efficiency, which is well explained by the fact that this nebula is illuminated by a B 9.5 star, whose cooler spectrum is less efficient to ionize PAHs (see Appendix C.4). The value of Re for Ced 201 indeed falls close to the model value of ϵ_PAH for a T_eff = 10 000 K, while a B 9.5 spectral type has T_eff ~ 11000 K. The lowest efficiencies are observed in protoplanetary disks (Fig. 9). In this case too there is a good agreement between Re and the modeled value of ϵ_PAH, which are low due to both the high γ values and cool spectral types (A and B) for these sources. Table 4 gives the values derived for ϵ_PAH for diffuse ISM conditions, that is T = 80 K and n_e = 3 × 10⁻² cm⁻³ for three classical radiation fields. Observations yield R_e = 2.1% (Table 2 and Appendix B), close but about a factor of ~2–3 lower than the model values, that is ϵ_PAH = 3.7, 5.9 and 3.9% for the interstellar radiation fields of Draine (1978), Mathis et al. (1983) and Habing (1968), respectively.

5.2.2 Heating rates

We compute the heating rates in sources for which there is a good estimate of the column density of the cooling gas, that is the Orion Bar (Salgado et al. 2016) and a diffuse ISM line of sight (Boulanger et al. 1996). For the Orion bar, with I_gas = 6.6 × 10⁻⁵ W m⁻² (Appendix B) and N_H = 7 × 10²² cm⁻² (Salgado et al. 2016), Λ_gas = 1.2 × 10⁻³⁰ W H⁻¹. This is close to the modeled gas heating rate by PAHs, that is Γ_PAH = 4.2 ± 2.0 × 10⁻³⁰ W H⁻¹, considering G₀ = 2 × 10⁴ (Joblin et al. 2018) and f_C = 0.01 (Castellanos et al. 2014) and the range of γ values in Table 2. For the diffuse ISM, Boulanger et al. (1998) find a cooling rate of Λ_gas = 3 × 10⁻³³ W H⁻¹ from FIRAS observations, in good agreement with the model values of Γ_PAH presented in Table 4.

6 Discussion

6.1 The role of PAHs in neutral gas heating

The comparison presented in Sect. 5 shows that there is a good agreement between the values of ePAH derived from the PAH heating model and the values of the observational diagnostics Re, for a wide range of physical conditions and environments. In addition, there is an excellent agreement in terms of the spatial variations of the charge state of PAHs in NGC 7023. The agreement between the modeled heating rates by PAHs and the gas cooling rates derived from the observations is also good for NGC 7023, the Orion Bar, and the diffuse ISM. Overall, this suggests that photoelectric heating by PAHs alone can explain the neutral gas heating in a number of environments. While this has been suggested by several studies (see Introduction), a direct assessment comparing the results from a molecular model including state-of-the-art molecular parameters for PAHs to observational diagnostics, had not been conducted before. BT94 pointed out the importance of PAHs, however 50% of the gas heating in their model is due to grains larger than N_C = 1500, while the present PAH model contains only PAHs of N_C = 54 C atoms. This stresses the central role that PAHs play in the physics of the gas heating of galaxies.

6.2 Heating by dust vs. heating by PAHs

Pabst et al. (2022) and earlier works have compared the theoretical photoelectric heating efficiency of dust from BT94, e, to $R_{\text{IR}}=\frac{I_{\text{gas}}}{I_{\text{IR}}}$, where I_IR is the total power radiated by dust in the far infrared. They found that ϵ > R_IR by a factor ~3 or more. This discrepancy is likely due to the fact that R_IR cannot easily be compared to ϵ, for two reasons. The first one is that, in the definition of BT94, e is computed by integrating the power absorbed (emitted) by dust in the 6–13.6 eV range, while some fraction of the observed radiated power IR is in reality due to absorption of UV photons with energies below 6 eV, and therefore IR overestimates the denominator value. Still, it appears unlikely that this effect only can explain this factor of 3 difference, because most of the energy absorbed by dust does lie above 6 eV. A most likely reason is that IR includes an important fraction of emission from dust grains which do not contribute at all to photoelectric heating.

6.3 PAH heating vs. X-rays, cosmic rays, shocks, and turbulence in neutral gas

Since the gas heating rate by PAHs Γ_PAH depends on their abundance (Eq. (29)), in regions where they are depleted, such as dense molecular clouds where PAHs are in condensed form (Rapacioli et al. 2006), other mechanisms are likely to take over, such as heating by cosmic rays (e.g. Padovani et al. 2009). In regions where the UV radiation field is weak compared to other sources of radiation such as X-rays (e.g. in the inner regions of protoplanetary disks near T Tauri stars, or near active galactic nuclei), heating by PAHs may also become negligible. In PDRs, heating by PAHs is likely to dominate, but heating by H₂ can also become important in dense regions near the dissociation front (Bron 2014). In some regions, shocks may become an important source of neutral gas heating. The competition between heating by PAHs and other sources of gas heating can be tested using Re: when this ratio is larger than the ePAH values predicted by the model, this is indicative that some other processes are at play. For instance, shock heating of neutral gas at the scale of galaxies results in extreme values of R_e (e.g. >0.3, Appleton et al. 2013, Table 1), well above the maximum efficiencies derived with our model (${\epsilon }_{\text{PAH}}^{\text{Max}}~0.11$, see Table 5). This illustrates how the PAH model efficiencies can be used as a diagnostic of UV radiative feedback vs. other sources of heating in galaxies.

7 Conclusion

In this article, we re-evaluate the contribution of PAHs to the photoelectric heating of neutral gas in astrophysical environments, using a simple analytical model in which state-of-the-art molecular parameters from laboratory measurements and quantum chemical calculations are included. We find that, for standard abundances of PAHs, the heating rates resulting from photoelectric heating by PAHs alone is comparable to those produced using a full size distribution of dust grains in the classical models of Bakes & Tielens (1994) and Weingartner & Draine (2001). The modeled and observed charge states of PAHs and their variation with physical conditions in NGC 7023 are in excellent agreement. The values of the photoeletric heating efficiency derived for PAHs from the model for a wide range of Galactic and extragalatic regions are in good agreement with the values of the observational diagnostics Re, that is the ratio of gas emission to the sum of gas + PAH emission. The values of the heating rate by PAHs derived from the model are also in good agreement with the observed gas cooling rates for the NGC 7023 nebula, the Orion Bar, and the diffuse ISM. Overall, this allows us to conclude that PAHs can explain most of the heating of the neutral gas in a variety of astrophysical environments, ranging from protoplanetary disks around young intermediate-mass stars to starburst galaxies in the early Universe. Our study highlights the importance to implement a robust description of the PE heating by PAHs in astrophysical codes computing the thermal balance of neutral gas, such as PDR models. The formalism presented here is included in a simple analytical model but it relies on our knowledge of the molecular properties of PAHs. This knowledge is crucial and laboratory studies (experiments and quantum chemical calculations) must be guided by astronomical observations. In particular, the coming spectroscopic data obtained from the MIRI and NIRspec spectrometers on board the James Webb Space Telescope will provide unprecedented details on the PAH populations (Berné et al. 2022).

Acknowledgements

O.B. acknowledges C. Pabst and J. R. Goicoechea for providing the data on the PE heating efficiency in Orion, J. Mc Kinney for providing the data regarding the PE heating efficiency in galaxies, F. Boulanger for fruitful discussions on the heating and cooling of the diffuse ISM, E. Josselin and A. Palacios for their help on extracting the stellar spectra used in this paper, and I. Schroetter for support on the writing of the code. The authors wish to thank L. Verstraete and V. Guillet for their comments on an early version of this work. We are grateful to the anonymous referee for his detailed report which greatly improved the manuscript. Finally, the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) ERC-2013-SyG, Grant agreement N^o610256 NANOCOSMOS. It has also been supported by the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES, and through an APR grant provided by CNES.

Appendix A Derivation of γ and R_e in NGC 7023

Appendix A.1 Derivation of γ in the NGC 7023 NW PDR

In the following, we describe how we derived values for the physical conditions, that is G₀, T and n_e, necessary to determine γ in NGC 7023, for 8 angular distances ranging from 20 to 60″ from the star. To derive the radiation field intensity, we used the procedure described in Pilleri et al. (2012) and adopted in Joblin et al. (2018). Briefly, this consists in using the sum of two identical synthetic spectra with effective temperatures 1.5×10⁴ K from the Kurucz (1993) library and a radius of 10 R_⊙ for the HD 200775 stars (Alecian et al. 2008). An additional extinction of A_V = 1.5 was applied to account for the presence of dust in the immediate surrounding of the stars. The radiation field intensity as a function of distance, in units of the Habing field, is derived assuming that the star-to-PDR inclination is of 63 degrees, which is necessary to reconcile the sky projected distance to the modeled star-to-PDR distance of 0.143 pc (Joblin et al. 2018). For angular distances from the star above 45″, the effect of dust absorption in the PDR is taken into account using a factor e^−τ(λ) where τ(λ) = σ_ext(λ)N_H, with N_H the total hydrogen column density along the axis between the PDR and the star. With a constant density in the PDR of n_H = 2 × 10⁴ cm⁻³ (Pilleri et al. 2012), the estimated values for NH at positions situated 50”, 55” et 60” from the star are 10²¹, 2 × 10²¹, and 3.2 × 10²¹ cm⁻², respectively. This corresponds to extinction values of A_V = 0.5,1, and 1.6 mag, respectively. The resulting radiation field intensities are computed by integrating the specific intensities in the 5.17 to 13.6 eV range, and are reported in Table 1. The gas temperature values for the positions up to 45” from HD 200775 are taken from the study of Boulais (2013) who modeled the spatial profile of the [C ii] 158 µm line emission using an LTE radiative transfer model to determine a stable temperature of ~ 750 K. The values adopted inside the PDR, that is above 45”, are from the models of Montillaud et al. (2013). Gas densities n_H in the cavity region up to 25” from the star are from Berné et al. (2015). The authors provided a value of n_H ~ 1.5 × 10⁴ cm⁻³ at ~ 25”, which we assume to be constant up to the PDR dissociation front from which we use a value of n_H ~ 2 × 10⁴ cm⁻³ (Pilleri et al. 2012). The gas temperature and density for all studied positions are reported in Table 1. The electron density is derived from the gas density by considering that all electrons originate from the ionization of carbon, that is n_e = 1.6 × 10⁻⁴n_H cm⁻³. The gas temperature and the electron density can then be used to derive the ionization parameter $\gamma =\frac{G_0\sqrt{T}}{n_{\text{e}}}$ (see values in Table 1). Andrews et al. (2016) used γ ~ 2 × 10⁴ K^1/2cm³ at the border of the PDR, (~ 42” from HD 200775), which is consistent with the values derived in the present study.

Appendix A.2 Derivation of R in the NGC 7023 NW PDR

We limit the study to the field of view (FoV) presented in Fig. 1, which excludes the parts of the nebula that are closer than 20” from the star, and farther than 60”. The former have to be excluded because other features attributed to species such as C60 and ${\text{C}}_{60}^{+}$ (Berné et al. 2013), and a strong thermal continuum due to hot dust grains (Berné et al. 2007) are present in the spectra. This contamination makes a clean analysis of the PAH emission with PAHTAT difficult. Similarly, at distances above 60”, the signal-to-noise ratio in the spectra becomes too low to use PAHTAT. In addition, regions too far from the star become optically thick in the mid IR (see Fig. 4 in Pilleri et al. 2012), which is also a challenge when using PAHTAT. We used the dataset obtained by Werner et al. (2004) with the IRS instrument aboard Spitzer. The data were previously analyzed by Berné et al. (2007) and Boersma et al. (2013), notably. The cube was obtained at a spatial resolution of 3.6” with a sampling of pixels of 1.8” and a spectral resolution of ~ 80, over a spectral range ranging from 5.5 to 15 µm. We applied the PAHTAT tool (Pilleri et al. 2012), which adjusts spectral templates of PAHs (notably cationic and neutral PAHs) to observations, to all spectra in the mid-IR cube to derive the integrated intensities of the PAH+ and PAH⁰ components (respectively $I_{{\text{PAH}}^{+}}$ and $I_{{\text{PAH}}^0}$) as well as their sum $I_{\text{PAH}}=I_{{\text{PAH}}^{+}}+I_{{\text{PAH}}^0}$. The R_i map is then obtained using $I_{{\text{PAH}}^{+}}$ and $I_{{\text{PAH}}^0}$ following Eq. 1.

Appendix A.3 Derivation of R_e in the NGC 7023 NW PDR

Gas cooling in the neutral ISM of galaxies is dominated by emission in fine-structure lines of [C ii] at 158 µm, and of [O i] at 63 µm and 145 µm. In addition, emission in the rotational lines of H₂, in the mid-IR, and emission in CO lines and other molecular species in the far IR can contribute. Bernard-Salas et al. (2015) provided a detailed study of gas cooling in the far IR in NGC 7023 with Herschel-PACS and SPIRE. We use their data for the [C ii] line at 158 µm, and the [O i] lines at 63 and 145 µm (Fig. A.1). We complement them with the 0-0 S(1), S(2), and S(3) rotational lines of H₂ at 17.0, 12.3, and 9.7 µm derived from Spitzer observations (Fig. A.1). The emission maps for the S(2) and S(3) H₂ lines were derived using PAHTAT (Pilleri et al. 2012) on the Spitzer-IRS SL data cube of the NGC 7023 NW PDR obtained by Werner et al. (2004). The map of the S(1) line integrated intensity was extracted using the CUBISM software (Smith et al. 2007a) on the LL data cube obtained by Werner et al. (2004). Overall, the gas line emission is largely dominated by the [O i] and [C ii] lines. H₂ emission is important only at the dissociation front (up to 30% maximum of the total line emission), but overall it represents less than 10% of the gas cooling. In addition, cooling by H₂ emission occurs in regions where gas heating is not dominated by PE heating, but rather by the formation of H₂, and the collisional de-excitation of vibrationally pumped H₂. Hence, we do not include H₂ emission in the gas cooling budget, gas being then the sum of [O i] and [C ii] lines emission. As for the derivation of the map of R_i, we limit the study to the FoV presented in Fig. 1, which excludes the parts of the nebula that are closer than 20” from the star, and farther than 60”. The Re map is then obtained from PAH (Sect. A.2) and gas following Eq. 2.

Appendix B Derivation of R_e and γ in a sample of sources

Appendix B.1 Galactic photodissociation regions

We include data for the Orion Veil, which is the extended nebular region in front and south of the M42 region of Orion A, illuminated by the O6 star Θ¹ Ori-C. It is a diffuse nebula where cooling is largely dominated by [CII] emission (Pabst et al. 2022). All the data used here were kindly provided by C. Pabst. It consists in the table of values of γ obtained for this region totaling over 11 000 pixels, as well as the values of the [CII] line intensities [CII] derived from SOFIA observations. In addition, we use the Spitzer-IRAC 8 µm maps of the region provided by Megeath et al. (2015). Emission in this IRAC filter (I_{8 µm}) is largely dominated by PAH emission, hence, in the absence of spectroscopic measurements, it can be used to evaluate I_PAH. We have used the data of NGC 7023, where I_PAH is obtained from spectroscopic analysis (Sect. A.2) as well as the I_8,µm IRAC maps, and found I_PAH ≈ 1.15 × 10⁻⁷I_{8 µm}, with I_PAH in W m⁻² sr⁻¹ and I_{8 µm} in MJy sr⁻¹. This is used to derive I_PAH from the IRAC data, and R_e is then given by $\frac{I_{\left[\text{CII}\right]}}{I_{\text{PAH}}+I_{\left[\text{CII}\right]}}$. The values presented in Fig. 3 and summarized in Table 2 are obtained by determining the median and first and 99^th percentiles for R_e, and median and 25^th and 75^th percentiles for γ. The used interval for γ is smaller than for R_e because of the presence of a larger number of outlier values of γ.

Fig. A.1 Emission maps of gas cooling. Maps on the upper row show the [C ii] line at 158 µm, as well as the [O i] lines at 145 µm and 63 µm, which were observed with Herschel-PACS at a resolution of 11”, 8.8”, and 4.5” respectively (Bernard-Salas et al. 2015). Maps on the lower row show the emission from the H2 rotational S(1), S(2) et S(3) lines at 17.035, 12.278, and 9.7 µm, respectively. These maps were extracted from the Spitzer archival data using CUBISM Smith et al. (2007a). The S(2) and S(3) maps are from the SL module of IRS (resolution of 3.6”), and the S(1) map is from the IRS LL module (resolution of 10”). Emission from the H2 v=1-0 S(1) line at 2.121 µm is presented in contours as a spatial reference for the dissociation front, but it is not included in the cooling budget, since this line is pumped by UV photons and hence does not cool the gas.

We include data for the Orion Bar (also illuminated by Θ¹ Ori-C) from Bernard-Salas et al. (2012) for the cooling lines ([OII], [CII]) at position (α = 5:35:19.778, δ = −5:25:30.65). We derive a total intensity for the cooling lines (dominated by [OI] at 63 µm) of 6.6 × 10⁻⁵ Wm⁻² sr⁻¹. We extract the value of I_8µm at this position both from the IRAC data and available IRS spectroscopy, which respectively yields I_8µm = 5500 and I_8µm = 7850 MJy sr⁻¹, which gives I_PAH = [6.32 – 9.02] × 10⁻⁴ Wm⁻² sr⁻¹. R_e is found to fall in the range [0.07–0.11]. The range of γ values is derived from model results for the Orion Bar presented in Joblin et al. (2018), and corresponding to the range of AV, for which the calculated [OI] and [CII] emissions are the dominant cooling lines. Values of R_e and γ for the Orion Bar are summarized in Table 2.

We include the IC 63 nebula, which is illuminated by the B0.5 star γ-Cas, and for this source I_PAH is taken from Table 3 in Pilleri et al. (2012). The values for the cooling lines are taken from Andrews et al. (2018). The error on absolute intensities of the cooling lines is ~ 30%, including uncertainty on beam filling factor and instrumental error. The values of γ are from Pabst et al. (2022).

Ced 201 is a nebula formed by a runaway B9.5 star penetrating a molecular cloud in Cepheus. For this source, PAH is also taken from Pilleri et al. (2012). The values for the cooling lines intensities are directly taken from the observations available in the Herschel Science Archive and extracted on the central spaxel of the PACS instrument. The same error as for IC 63 on the intensity, that is 30%, is considered. The values of γ are from Pabst et al. (2022).

The Horsehead is illuminated by the σ-Ori AB stars of spectral type O9.5V and B0.5V. The values for I_PAH are also taken from Pilleri et al. (2012), and the values for the cooling lines are also from the observations available in the Herschel Science Archive and extracted in the central spaxel of PACS (error = 30%). To estimate the values of γ we use a FUV radiation field of G₀ = 100, a temperature of T = 100 – 500 K, and a gas density of n_H = 1 − 20 × 10⁴ cm⁻³ (Habart et al. 2005). This yields γ values in the range [30–1400].

Appendix B.2 The diffuse interstellar medium

For the diffuse interstellar medium, we use the IR emission intensity for PAHs deduced by Compiègne et al. (2011), that is I_PAH = 1.45 × 10⁻³¹WH⁻¹ and the cooling measured for the [CII] line from Boulanger et al. (1996), that is Λ_gas = 3 ± 0.4 × 10⁻³³W H⁻¹, which yields R_e = 0.021 ± 0.002. We neglect the cooling from [OI] which is marginal for this low density medium. The average gas density ranges between 50 and 60 cm⁻³ (Boulanger 1999), and the gas temperature is T ~ 80 K. The radiation field in the diffuse ISM is G₀ ~ 1.7. With n_e = 1.6 × 10⁻⁴nH, this yields γ ∈ [1580 − 1900].

Appendix B.3 The Large Magellanic Cloud

For the Large Magellanic Cloud (LMC), we rely on the work of Lebouteiller et al. (2012) who provide a value of R_e = 0.07, and quote a factor of 2 uncertainty. We thus adopt R_e ∈ [0.05 − 0.01]. From Fig. 10 of Rubin et al. (2009), we derive G₀/n_e ∈ [630 − 1260], which for a temperature of 75 K (Rubin et al. 2009) yields γ ∈ [5460 − 10 900].

Appendix B.4 Star-forming galaxies

For star-forming galaxies in the local Universe, we use the ratio of the [CII] emission to the integrated flux in the 5 to 10 µm range provided in Helou et al. (2001). Following McKinney et al. (2021), we apply a factor 0.4 to this ratio to obtain Re, to account for the fact that PAH emission is responsible for only about 40% of the mid-IR emission of galaxies (Helou et al. 2001). The range of values of γ for the galaxies are taken from McKinney et al. (2021) using the same approach as for ULIRGs, which is based on the relationship of Díaz-Santos et al. (2017) and presented in the following section.

Appendix B.5 Ultra luminous infrared galaxies

McKinney et al. (2021) present measurements of the ratio between the PAH to [CII] emissions, which we adopt to derive Re. These authors also provide the values of G0/nH in a large sample of ultra luminous infrared galaxies (ULIRGs). Using these data, kindly provided in electronic format by the authors, we derive the median values of Re and γ and 25^th and 75 confidence intervals (see values in Table 2). We compute γ from G₀/n_H considering a temperature of 100 K, in agreement with estimated values for T in these galaxies (Díaz-Santos et al. 2017), and using n_e = 1.6 × 10⁻⁴ n_H.

Appendix B.6 High-redshift galaxy GS IRS20

McKinney et al. (2020) provide a measurement of the ratio of the [CII] cooling line to the intensity of the PAH 6.2 µm band in the GS IRS20 galaxy at a redshift z ≈ 1.9. This value and associated error bar can be converted to R_e using the calibration factor of 0.11 obtained by Smith et al. (2007b) on a large sample of nearby galaxies for the ratio of the 6.2 µm band to total PAH emission. Additional uncertainty stems from the lack of precise measurement of the 11.2 µm feature which implies a possible overestimation of the total PAH emission, of a factor of ~ 2 (McKinney priv. com.), which we include to obtain the maximum value of R_e. The range of values for γ is obtained by using the observed value of the IR surface density of this galaxy given in McKinney et al. (2020), that is ∑_IR ≈ 5 × 10¹¹ L_⊙kpc⁻², converted to a value of G0/nH using the law calibrated on ULIRGs by Díaz-Santos et al. (2017), and finally to a value of γ using a temperature of 100 K and n_e = 1.6 × 10⁻⁴ n_H.

Appendix B.7 Protoplanetary Disks

The disks for which we derived R_e are Herbig Ae/Be objects, which show prominent PAH emission. Amongst the disks observed by ISO and presented in Acke & van den Ancker (2004), we selected those for which there are Herschel measurements of the gas cooling lines (Meeus et al. 2012). This represents a total of 10 objects, which are listed in Table B.1. For some objects the [OI] 145 µm and [CII] lines were not detected. However this is not an issue since the [OI] 63 µm line is by far the most important cooling line in these objects, where the gas is dense and highly irradiated. The point presented in Fig. 3 represents the median value of Re from Table B.1 and the 25^th and 75^th percentiles. The range of values for γ can only be obtained indirectly. We used the results of the detailed model for PAHs in Herbig Ae/Be stars developed by Visser et al. (2007). In particular, these authors show that over 80% of the PAH emission arises from regions beyond the inner 100 AU of the disk, and their Fig. 5 provides the values for G₀/n_e, which range between 10³ and 10⁴ in this region of the disk. For a gas temperature of ~ 1000 K, this implies values for γ in the range [3 × 10⁴ − 3 × 10⁶]. The used temperature value is a rough estimate, however this choice is not critical since γ depends only on $\sqrt{T}$

Appendix C Derivation of the ionization potentials, ionization yields and partition coefficient

Appendix C.1 Ionization potentials

Following Wenzel et al. (2020), the values of the ionization potentials as a function of the charge Z, for Z ≥ 0, are given by:

$$I{P}^{\left(Z+1\right)+}=3.9\text{eV+}\frac{1}{4\pi {\epsilon }_0}\left[\left(Z+\frac{1}{2}\right)\frac{e^2}{a}+\left(Z+2\right)\frac{e^2}{a}\frac{0.03\text{nm}}{a}\right]\frac{1\text{C}}{e}\text{eV,}$$(C.1)

where ϵ₀ is the vacuum permittivity, e the elementary charge, and a the diameter of the molecule. The change of variable between a and the number of carbon atoms is given by (WD01):

$$a=\sqrt[3]{\frac{N_{\text{C}}}{468}}\text{nm}.$$(C.2)

For Z = −1, we use IP = 0 (but see Appendix C.3).

Appendix C.2 Ionization yields

Several studies provide measurements of ionization yields for neutral PAHs (Jochims et al. 1996; Verstraete et al. 1990) and cationic PAHs (Zhen et al. 2016; Wenzel et al. 2020). From these studies, empirical laws have been derived to model the ionization yields. For neutral species (Z=0), we use the law derived by Jochims et al. (1996) (energy values in eV):

$$Y\left(E\right)=\{\begin{array}{lll}\begin{array}{c}\frac{E-I{P}^{+}}{9.2}\\ 1\end{array}\hfill & \text{for}\hfill & \begin{array}{c}E\le {\text{IP}}^{+}+9.2\\ E>{\text{IP}}^{+}+9.2\end{array}\hfill \end{array}$$(C.3)

For the second ionization (Z = 1), we use the law provided by Wenzel et al. (2020):

$$Y^{+}\left(N_C,E\right)=\{\begin{array}{lll}\begin{array}{c}0\\ \frac{\alpha }{11.3-{\text{IP}}_{2+}}\left(E-{\text{IP}}^{2+}\right)\\ \begin{array}{l}\alpha \\ \frac{\beta \left(N_C\right)-\alpha }{2.1}\left(E-12.9\right)+\alpha \end{array}\\ \beta \left(N_C\right)\end{array}\hfill & \text{for}\hfill & \begin{array}{c}E<{\text{IP}}^{2+}\\ {\text{IP}}^{2+}\le E<11.3\\ 11.3\le E<12.9\\ \begin{array}{c}12.9\le E<15.0\\ E\ge 15.0,\end{array}\end{array}\hfill \end{array}$$(C.4)

where α = 0.3 and β is given by:

$$\beta \left(N_{\text{C}}\right)=\{\begin{array}{lll}\begin{array}{c}0.59+8.1·{10}^{-3}{N}_{\text{C}}\\ 1\end{array}\hfill & \text{for}\hfill & \begin{array}{c}32\le N_{\text{C}}<50\\ N_{\text{C}}\ge 50.\end{array}\hfill \end{array}$$(C.5)

Finally, for anions (Z=−1), we use Y = 1 following Visser et al. (2007).

Appendix C.3 Threshold energy for electron detachment

In Sect. 3.1, we have assumed that Y⁻(E) = 1 at all energies. This approximation can be justified by the low values of both the electronic affinity and the effective photoabsorption cross section at low energies (see Fig. 5). Recently, Iida et al. (2021) have shown that electron detachment from the pentacene anion, $\left({\text{C}}_{22}{\text{H}}_{14}^{-}\right)$, does not occur at the energy given by the electron affinity but is shifted to higher energies due to competition with radiative cooling. The authors estimated the survival rate of ${\text{C}}_{22}{\text{H}}_{14}^{-}$ to be 50% for an internal energy of 2.5 eV and 0% above 2.8 eV. We calculated the corresponding microcanonical temperature to be 775 and 824 K for 2.5 and 2.8 eV, respectively. The calculations use the density of states derived from a modified version of the Beyer-Swinehart algorithm (Mulas et al. 2006) and the harmonic vibrational frequencies from the Theoretical PAH Spectral Database. The electron affinity of cirumcoronene is calculated to be similar to that of pentacene (see Theoretical PAH Spectral Database). We therefore assumed that efficient electron detachment would occur for temperatures higher than 800 K, which correspond to internal energies higher than 6 eV for ${\text{C}}_{54}{\text{H}}_{18}^{-}$. The calculations were performed with the same method as for ${\text{C}}_{22}{\text{H}}_{14}^{-}$ and using the harmonic frequencies from the NASA Ames PAH IR Spectroscopic Database (Bauschlicher et al. 2018; Ricca et al. 2012).

In Fig. C.1, we compare the model results for the PAH heating rate r_PAH and the photoelectric heating efficiency of e_PAH as a function of the ionization parameter γ, which were obtained using Y⁻(E) = 1 for E ≥ 6 eV (and Y⁻(E) = 0 for E < 6 eV) compared to the case in which Y⁻(E) = 1 for all energies. The values of the heating rate and efficiencies are lower by a factor ~ 1.5 at γ = 10, by a few percent at γ = 100, and are not affected at γ = 100 and above. This indicates that the choice of threshold for Y⁻(E) does not have a significant effect on the results of the model, especially for the large values of γ that correspond to the objects studied in this paper (Fig. 9).

Appendix C.4 Partition coefficient

A value of 〈γ_e(E)〉 = 0.5 has been adopted in the literature (Verstraete et al. 1990; Tielens 2005; Bakes & Tielens 1994), based on the photoelectron kinetic spectra obtained for specific incident photon energies on benzene by Terenin & Vilessov (1964). We provide a revised estimate for this parameter based on the photoelectron spectroscopy measurements presented for the coronene molecule in Bréchignac et al. (2014). In particular, these authors provide (see their Fig. 2), for a given photon energy E absorbed in the range from 7.3 eV (IP of coronene) to 10.5 eV, the probability density function of kinetic energy of the photoelectrons E_K(E), from which $\langle \frac{E_K\left(E\right)}{E-IP}\rangle$ can be derived. Figure C.2 shows the values of γ_e(E) for coronene, from which 〈γ_e(E)〉 is obtained by averaging over energies, yielding 〈γ_e(E)〉 = 0.46 ± 0.06 (68% confidence interval), in agreement with the classical value of 0.5.

Fig. C.1 Model derived heating rate ΓPAH (upper panel) and pah photoelectric heating efficiency ϵPAH (lower panel) obtained for IP0 = 0 and 6 eV. In both cases, the model is computed with Nc = 54, T = 500 K, and TEff = 3 × 104 K. For the heating rate, we use fC = 0.1.

Fig. C.2 Fraction of the energy that goes into kinetic energy of the photoelectron, γe(E), as a function of incident photon energy, derived from the photoelectron spectroscopic measurements of Bréchignac et al. (2014) for coronene, C24H12. The sharp features are induced by the presence of auto-ionization states.

References

All Tables

All Figures

	Fig. 1 General view of the NGC 7023 NW PDR. Grey image is Spìtzer-IRAC data at 8 µm. Green contours show the emission in the H2 1–0 S(1) line at 0.5, 1 and 2 × 10−7 W m−2 sr−1, from observations at CFHT presented in Lemaire et al. (1996). The orange contour defines the region considered here for the PE heating study.
In the text

	Fig. 2 Maps of the ionization fraction of PAHs in NGC 7023 (upper panel), and ratio of gas cooling to PAH emission (lower panel).
In the text

	Fig. 3 Measured emission ratio Re (Eq. (2)), as a function of physical conditions traced by the ionization parameter y, in a sample of astro-physical objects.
In the text

	Fig. 4 Ionization yields YZ(E) for charge states Z = 0 (solid lines) and Z = 1 (dashed lines), computed from Eqs. (C.3)–(C.5) respectively, and for the three different PAH sizes. The yield for Z = −1, Y−(E) = 1 is not shown, for clarity of the figure.
In the text

	Fig. 5 Average photoabsorption (σ, solid line) and photoionization (σion, dashed line) cross sections per C atom of PAHs adopted in the 4 energy level model.
In the text

	Fig. 6 Fractions in the different PAH charge state Z =−1,0,1,2 as a function of the ionization parameter y, which were derived using our PAH model and Teff = 3 × 104 K.
In the text

	Fig. 7 Photoelectric efficiency ϵPAH computed for NC = 54, Teff = 30 000 K, and T = 100 and 1000 K with the PAH heating model. For comparison, we give the photoelectric efficiency ϵΓ computed by WDOl for grains with a size a = 4.9 A equivalent to NC = 54 (Eq. (C.2)), and the heating efficiency for PAHs given in Tielens (2021).
In the text

	Fig. 8 Photoelectric heating rate of the gas. Dash-dotted line: average model from Weingartner & Draine (2001; see text for details). Dashed line: model from Bakes & Tielens (1994). Continuous lines: PAH model (this work) for two PAH abundances, that is fC = 0.05,0.1.
In the text

	Fig. 9 Observed emission ratio Re (Table 2) and modeled photoelectric heating efficiency for PAHs, ϵPAH, as a function of the ionization parameter γ, for various stellar effective temperatures. The upper curve corresponds to Teff = 4 × 104 K, and the lowest curve to Teff = 104 K. All models use NC = 54 and a gas temperature of T = 500 K.
In the text

	Fig. 10 Variation with γ of the PAH ionization fraction Ri in NGC 7023 (diamonds with errobars). Values derived from the map in Fig. 2 for the eight angular distances between 20 and 55″ from HD 200775 given in Table 1. The values of fi from the PAH model are shown with lines for a gas temperature T = 500 K, and three PAH sizes.
In the text

	Fig. 11 Variation with y of the emission ratio Re in NGC 7023 derived from the map in Fig. 2 for the eight angular distances between 20 and 55″ from HD 200775 given in Table 1. The values of ϵPAH from the PAH model are shown with lines for a gas temperature T = 500 K, and three PAH sizes.
In the text

Fig. A.1 Emission maps of gas cooling. Maps on the upper row show the [C ii] line at 158 µm, as well as the [O i] lines at 145 µm and 63 µm, which were observed with Herschel-PACS at a resolution of 11”, 8.8”, and 4.5” respectively (Bernard-Salas et al. 2015). Maps on the lower row show the emission from the H2 rotational S(1), S(2) et S(3) lines at 17.035, 12.278, and 9.7 µm, respectively. These maps were extracted from the Spitzer archival data using CUBISM Smith et al. (2007a). The S(2) and S(3) maps are from the SL module of IRS (resolution of 3.6”), and the S(1) map is from the IRS LL module (resolution of 10”). Emission from the H2 v=1-0 S(1) line at 2.121 µm is presented in contours as a spatial reference for the dissociation front, but it is not included in the cooling budget, since this line is pumped by UV photons and hence does not cool the gas.

In the text

	Fig. C.1 Model derived heating rate ΓPAH (upper panel) and pah photoelectric heating efficiency ϵPAH (lower panel) obtained for IP0 = 0 and 6 eV. In both cases, the model is computed with Nc = 54, T = 500 K, and TEff = 3 × 104 K. For the heating rate, we use fC = 0.1.
In the text

	Fig. C.2 Fraction of the energy that goes into kinetic energy of the photoelectron, γe(E), as a function of incident photon energy, derived from the photoelectron spectroscopic measurements of Bréchignac et al. (2014) for coronene, C24H12. The sharp features are induced by the presence of auto-ionization states.
In the text