© T. Schirmer et al. 2022

1 Introduction

Interstellar dust is ubiquitous in the interstellar medium (ISM) and it is involved into the physical, chemical, and dynamical evolution of numerous environments through different processes such as the gas heating through the photoelectric effect (Bakes & Tielens 1994; Weingartner & Draine 2001) and the H₂ formation on dust surfaces (e.g Le Bourlot et al. 2012; Bron 2014; Jones & Habart 2015). The efficiency of these processes depends crucially on the dust properties (size, composition, and shape). It is therefore crucial to constrain those properties in order to understand better the different environments where dust exists. However, the broad disparity in the physical conditions (density and irradiation) triggers an evolution of these dust properties through grain growth (i.e. accretion and coagulation), grain destruction (i.e. photo-destruction and collisions), and processing (i.e. aromatisation, dehydrogenation), which are not yet fully understood.

The radiative feedback of freshly formed stars irradiating their nearby dense environments leads to the creation of the well-known photon-dominated regions (PDRs). In these regions, physical conditions vary significantly on small spatial scales which is why PDRs are a unique place to study how dust evolves as a response to the physical conditions. The mid-IR spectra of PDRs present a wealth of emission band features due to the smallest grains overlying the continuum of hot dust emission which has been extensively analysed using the Infrared Space Observatory (ISO) and Spitzer data. Strong variations in the spectra have been found across and between PDRs (e.g. Peeters et al. 2002, 2004; Rapacioli et al. 2006; Abergel et al. 2002; Berné et al. 2007). With Herschel data in the far-IR (FIR), it has also become possible to study the emission of large grains in thermal equilibrium (e.g. Abergel et al. 2010; Arab et al. 2012). Using the dust model THEMIS (Jones et al. 2013, 2017) together with the 3D radiative transfer code SOC (Juvela 2019), the study from Schirmer et al. (2020, hereafter called Paper I) reported that the nano-grain dust-to-gas mass ratio in the irradiated outer part of the Horsehead is 6–10 times lower than in the diffuse ISM and that the minimum size of these grains is 2–2.25 times larger than in the diffuse ISM.

The gas physics and chemistry in PDRs are strongly affected by variations in the dust properties (Schirmer et al. 2021). The aim of this study is thus to constrain nano-grain dust properties in the Orion Bar and in IC63, two PDRs that present very contrasted physical conditions, using Spitzer and Herschel observations. Evidence of dust evolution in those two PDRs has already been demonstrated (e.g. Arab et al. 2012; Van De Putte et al. 2019) and our goal in this work is to understand the different mechanisms that lead to the formation and destruction of nano-grains in these regions.

The paper is organised as follows. In Sect. 2, we describe the previous studies of the Orion Bar and IC63. We also present the Spitzer and Herschel observations. In Sect. 3, we detail the THEMIS dust model as well as the radiative transfer code used to compute dust emission. In Sect.4, we present the density profiles and our methodology to constrain those profiles. In Sect. 5, we compare the dust modelled emission to the observations and the constraints on the dust properties that result from this modelling. In Sect. 6, we discuss our results and present a scenario of dust evolution within these regions, based on processing timescales. Our conclusions are given in Sect. 7.

Fig. 1 Selected PDRs seen with Spitzer. Top: IC63 seen at 3.6 µm. Bottom: Orion Bar seen at 3.6 µm. The white solid lines correspond to the cuts used in our study.

2 Selected PDRs

2.1 Orion Bar

The Orion Bar (see Fig. 1, bottom panel) is a bright filament of the Orion molecular cloud, a site located at 1 pc (Menten et al. 2007) that is undergoing massive star-formation. The bar is illuminated by the O7-type star θ¹ Ori C, the most massive member of the Trapezium young stellar cluster, at the heart of the Orion Nebula (about 2′ north east of the Bar, e.g., O’Dell 2001). The intense ionizing radiation and strong winds from θ Ori C power and shape the nebula (Pabst et al. 2019; Güdel et al. 2008), a blister Hii region that is eating its way into the background parental cloud Orion Molecular Cloud (OMC). The incident UV radiation field is (1−3) ×10⁴ times the mean interstellar field (e.g., Marconi et al. 1998). The first PDR layers are predominantly neutral and atomic: [H] > [H₂ ] ≫ [H⁺]. They display a wealth of near-infrared (NIR) atomic lines from low ionization potential elements (forbidden lines, recombination lines, etc.; see Walmsley et al. 2000). Gas is mainly heated by photoelectrons ejected from small grains and it is mostly cooled by the FIR [Cii] 158 µm and [Oi] 63 µm fine-structure lines (e.g. Tielens et al. 1993; Herrmann et al. 1997; Bernard-Salas et al. 2012; Ossenkopf et al. 2013). The observed narrow (Δν = 2–3 km s⁻¹) carbon and sulfur radio recombination lines also arise from these layers (not from the Hii region) and provide a measure of the electron density in the PDR (n_e ≃ 10−100 cm⁻³; e.g. Wyrowski et al. 1997; Cuadrado et al. 2019; Goicoechea et al. 2021). The atomic PDR zone also hosts the peak of the mid-infrared (MIR) polycyclic aromatic hydrocarbons (PAH) emission (e.g. Bregman et al. 1989; Sellgren et al. 1990; Tielens et al. 1993; Giard et al. 1994; Knight et al. 2021), which has led to numerous studies on dust in the Orion Bar. Indeed, based on ISO spectroscopy with the SWS spectrograph of the Orion Bar, Peeters et al. (2002) showed that the variations in the spectral features in the 6−9 µm spectral range (emission bands at 6.0, 6.2, 6.6, 7.0, 7.7, 8.3, and 8.6 µm) are linked to variations in the local physical conditions but also with both formation and evolution processes associated with PAHs. From imaging and spectroscopic observations with ISOCAM (5−16 µm), Abergel et al. (2002) showed that at the illuminated edge of the Orion Bar, there is a systematic decrease in the intensity of the aromatic features (especially at 7.7 µm) relative to the continuum inside the shielded molecular regions compared to the photo-dissociated and photo-ionised regions. Later, Arab et al. (2012) showed that a decrease in the PAH abundance together with an increase in the larger grain emissivity, fits the Spitzer and Herschel observations of the Orion Bar well. This can be explained via the scenario of PAHs being photodestroyed or coagulating onto the surface of larger grains, thus increasing their emissivity. Mid-IR photometry with SOFIA of the Orion Bar and the Orion HII region (Salgado et al. 2016) requires coagulated grains to explain the decrease by a factor of 5–10 of the UV and infrared dust opacities from the diffuse ISM to these PDRs. Using SOFIA, Spitzer, UKIRT, and ISO observations together with the dust destruction model SHIVA (Murga et al. 2019), Murga et al. (2021) showed that small PAHs (N_C < 60) are most likely destroyed.

From now on, we follow Arab et al. (2012) and adopt G₀ = 2.6 × 10⁴ where:

$$G_0=\frac{1}{1.6\times {10}^{-3}\left(\text{erg\hspace{0.17em}}{\text{s}}^{-1}\right)\left({\text{cm}}^{-2}\right)}{\displaystyle {\int }_{\text{6eV}}^{13.\text{6eV}}{I}_v\text{d}v.}$$(1)

2.2 IC63

Since it was first discovered with the telescope of the Mount Wilson and Palomar Observatories then mentioned in Sharpless (1953), IC63 (see Fig. 1, top panel) has been the subject of various studies (e.g. Fleming et al. 2010; Andersson et al. 2013; Andrews et al. 2018; Dennis 2020; Lai et al. 2020; Soam et al. 2021a,b). At the very beginning, Witt et al. (1989) found evidence of Extended Red Emission (ERE) in IC63, based on low-resolution UV (115−195 nm) spectra obtained with the International Ultraviolet Explorer (IUE) together with spectra obtained at the McGraw-Hill Observatory, which covers a spectral range from 500 to 900 nm. They estimated that the UV field hitting the surface of IC63 is about ~2.3 × 10⁵ photons cm s Hz⁻¹ at 100 nm, hence¹ χ ~ 680 (i.e. G₀ ~ 1160). Using this χ value together with the IUE observations, Sternberg (1989) found an average density of n ~ 4 × 10⁴ H cm⁻³ in this nebula. Based on millimeter and submillimeter observations², Jansen et al. (1994) constrained the UV field as well as the average density in IC63 using HCO⁺, CS, HCN, and H₂CO lines. They found that χ is about 650 (i.e. G₀ ~ 1110) and n(H₂) ~ (5 ± 2) × 10⁴ cm⁻³ (n_H ~ 2 n(H₂) ~ 1 ± 0.4 × 10⁵ H cm⁻³). These results where later reinforced on the basis of chemical calculations for various physical conditions with n_H varying from 6 × 10⁴ and 14 × 10⁴ H cm⁻³, Jansen et al. (1995) found a best fit corresponding to n_H = 1 × 10⁵ H cm⁻³. This value was confirmed in Jansen et al. (1996) and they showed that the carbon abundance in gas-phase is $X_{\text{C}}=\left({13}_{-4}^{+6}\right)$% although it is about 30–60% in diffuse clouds such as ζ Oph (Cardelli et al. 1993). Using H₂ pure-rotational lines with the Short Wavelength Spectrometer (SWS) onboard the Infrared Space Observatory (ISO) together with major fine-structure cooling lines of OI at 63 µm and 145 µm as well as CII at 157.7 µm, Thi et al. (1999) found that the incident radiation field has a G₀ of about 10³, which falls in the range of the G₀ values found in previous studies. Observations at IRAM 30-m of C₂H, c-C₃H₂, C₄H, 1-C₃H, c-C₃H, and HN¹³C with relatively high abundance might support an in situ formation of carbon chains and rings assisted by a release of acetylene from very small carbon particles (Fossé et al. 2000; Teyssier et al. 2004) or the release of C_nH_m species from the photo-processing of a-C(:H) nano-particles (Jones & Habart 2015). Based on extinction mapping, evidence for dust evolution has been shown in IC63 (Van De Putte et al. 2019). As G₀ estimates vary between G₀ ~ 1000 (Thi et al. 1999) and G₀ ~ 1200 (Witt et al. 1989; Jansen et al. 1994), we chose G₀ = 1100.

2.3 Ancillary data used in this study

We use Spitzer and Herschel observations (see Fig. A.1) in six photometric bands (3.6, 4.5, 5.8, 8, 24, and 70 µm) for IC63 and in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm) for the Orion Bar³ (see Fig. A.1). The processing of the Spitzer maps is detailed in Bowler et al. (2009). We study the observed emission profiles through a cut across both of these PDRs (see solid white lines in Fig. 1). The calibration uncertainty in the IRAC bands (IRAC3.6, IRAC4.5, IRAC5.8, and IRAC8.0) is 2% (Reach et al. 2005), 4% in MIPS24 (Engelbracht et al. 2007), and 5% in PACS70 (Gordon et al. 2007). We considered all these errors to be independent of the wavelength to first order. As the minor contribution of the gas in those two bands does not affect the bulk of our results, we therefore consider that the observed emission is dust emission.

3 Dust populations in optically thick regions

Photon-dominated regions are optically thick and radiative transfer is therefore required to properly model dust emission. We used the 3D radiative transfer code SOC (Juvela 2019), together with the THEMIS dust model (described hereafter).

The Heterogeneous dust Evolution Model for Interstellar Solids⁴ (THEMIS, e.g., Jones et al. 2013, 2017) provides a description of grain properties that reflects evolutionary processes. In the ISM, dust evolution is mostly driven by UV photons and collisions between grains or with gas species. The efficiency of these processes, set by the UV flux G₀ and the density n_H, determines the structure and composition of the dust. This dust model is based on two main dust materials that are amorphous olivine-type and pyroxene-type silicates with iron and iron sulphide nano-inclusions a-Sil_Fe,FeS, and amorphous hydrocarbons solids a-C(:H) materials (see Jones 2012c,b,a), which encompasses a-C:H materials that are H-rich and aliphatic-rich and a-C materials that are H-poor and aromatic-rich.

Another major contribution from THEMIS is the core-mantle description of dust grains. For the purpose of understanding this depiction and especially in the context of PDRs, it is important to understand the influence of UV irradiation that chiefly affects the carbonaceous dust population. Such UV photons can photo-destroy C-H bonds and therefore, generate the creation of C=C bonds. Thus, the irradiation of a-C(:H) materials leads to their progressive aromatisation⁵. As the typical penetration depth of a UV photon in a-C(:H) material is about 20 nm (see Fig. 15, Jones 2012b), carbonaceous grains that are smaller than 20 nm are entirely photo-darkened and, hence, aromatic-rich, these are a-C grains. Regarding larger carbonaceous grains, they are composed of an aliphatic-rich a-C:H core surrounded by an H-poor and aromatic-rich a-C mantle assumed to be 20 nm thick, which prevents the photo-processing of the core, thus allowing it to remain aliphatic-rich. These are the large a-C:H/a-C grains. This view provides us with a continuous description of carbonaceous grains from the smallest, which contain aromatic cycles and are stochatiscally heated to the largest that are in thermal equilibrium. Regarding the silicates, it is assumed that this dust population is composed of a core of silicate, surrounded by a mantle of a-C.

The THEMIS model for the diffuse ISM is therefore composed of three dust populations that are built upon the two materials of carbon (a-C(:H)) and silicate (a-Sil) described above. These three dust populations are defined as follows:

• a-C(:H) dust population whose size distribution follows a power-law with an exponential cut-off. Since about 80% of the mass of this population is found in grains smaller than 20 nm, and thus mostly aromatic-rich, we refer to it as a-C grains or nano-grains indifferently in the following.

• a-C(:H) dust population whose size distribution follows a log-normal law. As this population is essentially composed of a-C:H/a-C core-mantle grains (99% of mass), we refer to it as a-C:H/a-C grains, although a few a-C are included.

• a-Sil/a-C dust population whose size distribution follows a log-normal law.

The size distribution of these three populations are shown in Fig. 2 and the associated parameters are detailed in Table. 1.

Fig. 2 Dust size distributions for a-C (solid line), a-C:H/a-C (dotted line), and a-Sil/a-C (dotted dashed line).

4 Constraining the PDR structure

Radiative transfer requires information regarding the gas density profile. As we lack information about the density profile across IC63, we used a method to constrain it based only on dust emission. From now on, we take the following analytical expression to describe the density profile across these PDRs:

$$n_{\text{H}}\left(z\right)\{\begin{array}{ll}{n}_0\times {\left(\frac{z}{z_0}\right)}^{\gamma }\hfill & \text{if\hspace{0.17em}}z<z_0\hfill \\ n_0\hfill & \text{if\hspace{0.17em}}z>z_0,\hfill \end{array}$$(2)

with z as the position from the edge of the PDR, γ the power-law exponent of the gas density profile, and z₀ the depth beyond which constant density n₀ is reached. This profile has already been used in PDR studies (e.g. Habart et al. 2005; Arab et al. 2012; Schirmer et al. 2020).

4.1 IC63

No previous study has established the density profile across IC63. However, based on the studies of Jansen et al. (1994) about HCO⁺, CS, HCN, and H₂CO gas lines and Jansen et al. (1995) on chemical calculations, we can set n₀ to 1 × 10⁵ H cm⁻³. Following the studies of Arab et al. (2012) on the Orion Bar and Schirmer et al. (2020) on the Horsehead, we set γ = 2.5.

In Appendix B, we show that compared to variations in z0, variations in the dust size distribution barely affect the width⁶ of the dust emission profiles. We therefore constrain z₀ through the comparison between the width of the dust modelled and observed emission profiles in the different photometric bands.

We computed dust emission across IC63 using THEMIS for z0 that varies from 0.001 to 0.020 pc on a linear grid of 100 points. We then compared the full width at half maximum (FWHM) of the observed and modelled dust emission profiles by minimising the following ${\chi }_{\text{wid}}^2$:

$${\chi }_{\text{wid}}^2={{\displaystyle \sum _{i\in \text{\hspace{0.17em}filters}}\left({\text{Δ}}_{i,\mathrm{mod}}-{\text{Δ}}_{i,obs}\right)}}^2,$$(3)

where Δ_i,mod (Δ_i,bos) is the FWHM of the modelled (observed) dust emission profiles in the i-th band. The FWHM of the observed dust emission profiles from 3.6 to 70 µm for both IC63 and the Orion Bar can be found in Table 2.

We show the results of this minimisation in Fig. 4 (left panels). Regardless of the band, the FWHM increases with z₀ (see Fig. 4, top left panel) which is expected as the larger z₀ the smoother the density profile slope hence the radiation can penetrate deeper in the cloud to heat dust grains, thus increasing the FWHM. Regarding the χ² minimisation (see Fig. 4, bottom left panel), there is a strong minimum for z₀ = 0.004 pc which we adopt in the following. To summarise, the parameters that describe the density profile across IC63 are:

$$\begin{array}{lll}{n}_0=1\times {10}^5{\text{Hcm}}^{-3};\hfill & z_0=0.004\text{pc;}\hfill & \gamma =\mathrm{2.5.}\hfill \end{array}$$(4)

4.2 Orion Bar

Conversely to IC63, many studies have been carried out with the aim of exploring the Orion Bar (see Sect. 2.1). In our study, we use the density profile from Arab et al. (2012). Whilst it provides a value of z₀ – in addition to n₀ and γ –, we choose to constrain it using the same procedure as for IC63.

We show the results of the χ² minimisation in Fig. 4 (right panels). There is a clear minimum for z₀ = 0.025 pc. One can note that this value is nearly twice as small as in the study of Arab et al. (2012). This discrepancy is probably due to the radiative transfer modelling. Indeed, Arab et al. (2012) used a 1D radiative transfer code whereas we use a 3D radiative transfer code. In their modelling, it is assumed that 88% of the photons are forward scattered (i.e. 12% are back scattered); however, in a 3D configuration, a more extensive treatment of the scattering will lead to less forward scattering photons. Nevertheless, those differences barely affect the dust properties that are obtained in both studies. To summarise, the parameters that describe the density profile across the Orion Bar are the following:

$$\begin{array}{lll}{n}_0=1.5\times {10}^5{\text{Hcm}}^{-3};\hfill & z_0=0.025\text{\hspace{0.17em}pc;}\hfill & \gamma =\mathrm{2.5.}\hfill \end{array}$$(5)

We note that z₀ is six times larger in the Orion Bar than in IC63. However, we cannot conclude that the density profile in situ is steeper in IC63 as it could also be due to spatial resolution effects.

5 Constraining dust properties

The approaches pursued to constrain the dust properties in IC63 and the Orion bar are slightly different due to the specificity of those two objects. In this section, we present those approaches and the constraints on the nano-grain properties obtained with the radiative transfer modelling. A summary of the results can be found in Table 3.

Fig. 3 Comparison between the observed dust emission and the modelled dust emission in IC63 using the best set of dust parameters (Ma_C/MH = 0.10 × 10−2, amin.a-C = 0.70 nm, and α = −5) in six photometric bands (3.6, 4.5, 5.8, 8, 24, and 70 µm). The dust modelled (observed) emission is shown in blue (green) line. The cut considered across IC63 is shown in Fig. A.1.

Fig. 4 ${\chi }_{\text{wid}}^2$ minimisation for IC63 and the Orion Bar. Top: FWHM of the observed dust emission profiles in different photometric bands as a function of zo (left for IC63 and right for the Orion Bar). Bottom: ${\chi }_{\text{wid}}^2$ as a function of z0 (left for IC63 and right for the Orion Bar). The vertical black dashed lines corresponds to the minimum value of ${\chi }_{\text{wid}}^2$. Radiative transfer has been done using diffuse ISM-like dust with an incident radiation field corresponding to a blackbody at 25 000 K (38 000 K) with G0 = 1100 (G0 = 2.6 × 104) for IC63 (Orion Bar).

5.1 Two objects, two approaches

Before going further, it is necessary to present the parameters associated with the nano-grain properties we aim to constrain. Three parameters associated with their size distribution are varied: (1) the abundance, that is, the a-C mass to gas ratio, M_{a_c}/Mh; (2) the minimum size, a_min,a-C; and (3) the slope of the power-law size distribution, α. The influence of variations in these parameters on both the dust size distribution and the associated spectra in the optically thin limit are shown in Paper I (Fig. 4, first line for the dust size distributions and second line for the associated spectra). Another important parameter is the length of the PDR along the line of sight, l_PDR. As this parameter does not affect the shape of the dust spectrum (see Sect. 4.2 in Paper I), l_PDR can be adjusted after the fact.

A simplified approach would be to explore the 3D space defined by M_a__C/M_H, a_{min, a-C}, and α and then computing the radiative transfer for each set of parameters. However, if we assume a subset of N values for each parameter, the computation time would be proportional to N³, which is time-consuming. In our case, as discussed below, it is possible to divide the exploration of the 3D space into a 2D exploration (a_{min, a-C}, α) and then follow up with a ID (M_{a_C}/M_H) exploration⁷.

In the case of the Orion Bar, the observations consist of four photometric bands in the mid-IR (MIR, i.e. 3.6, 4.5, 5.8, and 8 µm)and one in the NIR/FIR (70 µm). Variations in a_{min, a-C} and α change the dust spectral shape (i.e. ratios between those four bands evolve non-linearly) in the MIR but barely affect the FIR dust emission (see Fig. 4 in Paper I). On the contrary, variations in M_a__C/mh do not affect the spectral shape in the MIR but, instead, the MIR-to-FIR dust emission ratio⁸ only. In this specific case, we therefore adjust the shape of the MIR dust spectrum by exploring the 2D space (a_{min, a-C}, α) and then we adjust the MIR to FIR dust emission ratio by varying M_a__C/mh. To this end, we define:

$${\chi }_{\text{MIR}}^2={\displaystyle \sum _{i\in \text{\hspace{0.17em}filters}\left(\text{MIR}\right)}{\left(\frac{X_i-{\mu }_{\text{MIR}}}{{\sigma }_i}\right)}^2,}$$(6)

and

$${\chi }_{\text{tot}}^2={\displaystyle \sum _{i\in \text{\hspace{0.17em}filters}}{\left(\frac{X_i-{\mu }_{\text{tot}}}{{\sigma }_i}\right)}^2,}$$(7)

with

$$\begin{array}{lll}{X}_i=\frac{I_{\text{obs},\max }\left(i\right)}{I_{\text{mod},\max }\left(i\right)}\hfill & ;\hfill & {\sigma }_i=r_{\text{obs}}\left(i\right)X_i,\hfill \end{array}$$(8)

and

$$\begin{array}{ll}{\mu }_{\text{MIR}}={\langle X_i\rangle }_{i\in \text{\hspace{0.17em}filters}\left(\text{MIR}\right)};\hfill & {\mu }_{\text{tot}}={\langle X_i\rangle }_{i\in \text{\hspace{0.17em}filters}},\hfill \end{array}$$(9)

where r_obs is the relative error for each band and defined in Sect. 2.3 and I_obs,max(i) = max (I_obs,i(z)) with I_obs,i(z), the dust observed in the i-th band at the position z along the cut ${\chi }_{\text{MIR}}^2$ only takes into account the MIR bands and is therefore sensitive to variations in a_{min a-C} and α. Conversely, ${\chi }_{\text{MIR}}^2$ does not depend on M_a__C/mh. We therefore adjust the shape of the MIR dust spectrum by minimising ${\chi }_{\text{MIR}}^2$ in the 2D space (a_{min, a-C}, α) and then we adjust the overall MIR to FIR ratio by minimising ${\chi }_{\text{tot}}^2$.

In the case of IC63, it is not possible to apply the same method as variations in M_a__C/mh lead to a non-linear evolution of the ratio between the 24 µm band with each of the four bands in the MIR. We therefore have to minimise ${\chi }_{\text{tot}}^2$ in the 3D space (M_a__C/M_H, a_min,a-C, α).

Fig. 5 ${\chi }_{\text{tot}}^2$ in the 2D space (Ma_C/MH and amin, a-C) for α = −5 in the IC63 model.

5.2 IC63

We study the ${\chi }_{\text{tot}}^2$ distribution in the 3D space (M_{a_C}/M_H, a_min,a-C, and α), defined as follows:

1. M_a__C/M_H varies from 0.05 × 10⁻² to 0.017 × 10⁻² on a 10-step linear grid.

the width of the dust emission profiles barely depends on the dust properties and solely depends on the density profile. Except at 4.5 µm, the dust emission maxima are well reproduced. The discrepancy at 4.5 µm worsens as this band has been deliberately excluded from the ${\chi }_{\text{tot}}^2$ minimisation because we are not able to simultaneously fit the observations in all of the six photometric bands. This discrepancy has already been observed and discussed in more details in Sect. 6.1 of Paper I for the Horsehead Nebula.

Fig. 6 Top: Ma-C/MH and amin,a-C associated with the minimum ${\chi }_{\text{tot}}^2$ in the 2D space (Ma-C/MH, amin,a-C) as a function of α (see Fig. C.1). Bottom: minimum value of ${\chi }_{\text{tot}}^2$ in the 2D space (Ma-C/MH, amin,a-C) as a function of α.

5.3 Orion Bar

5.3.1 Contraining a_{min a–C} and α

We study the ${\chi }_{\text{MIR}}^2$ distribution in the 2D space (a_{min a–C} and α), defined as follows:

1. a_{min, a–C} varies from 0.35 to 0.9 nm on a 40-point linear grid;

2. α varies from −3 to −13 on a 40-point linear grid.

Figure 7 (left panel) shows the result. We observe a clear degeneracy between a_min,a–C and α, the first increasing when the second decreases. This is expected as a decrease in α implies an increase in the NIR/MIR dust emission, which is counterbalanced by an increase in a_min,a–C. We also see that the ${\chi }_{\text{MIR}}^2$ is lower than eight in the degeneracy, which means that the dust MIR spectrum is well reproduced. We choose five couples of a_min,a–C and α (see the symbols in Fig. 7, left panel) to cover the whole degeneracy and then constrain Ma-C/MH. Finally, it is important to note that despite the degeneracy α ≤ 6, which is linked to the fragmentation of large grains into nano-grains (see Sect. 6.3) and a_min,a–C ≤ 0.8 nm, which is linked to the photo-destruction of nano-grains (see Sect. 6.2).

5.3.2 Constrain M_a-C/M_H

We studied the ${\chi }_{\text{tot}}^2$ distribution in the 1D space (M_a–C/M_H), where M_a–C/M_H varies from 1 × 10⁻⁵ to 0.17 × 10⁻² on a logarithmic scale composed of 25 points. According to the previous section, we take five different couples of a_min,a–C and α (1: 0.35 nm/−5.96, 2: 0.51 nm/−5.96, 3: 0.63 nm/−7.44, 4: 0.71 nm/−8.92, 5: 0.77 nm/−12.25) that minimise ${\chi }_{\text{MTR}}^2$.

Figure 7 (right panel) shows the results. There is a minimum value for each couple a_min which provides a range of M_a–C/M_H values from ~1 × 10⁻⁵ to ~4 × 10⁻⁵. To summarise, while there is a degeneracy between the best amin, a-C and α values, we are able to provide a constraint on M_a-C/M_H. We show in Fig. 8 the comparison between the dust observed emission and the dust modelled emission for one of the five different set of best parameters. The same figures for the four other models can be found in Fig. A.1.

Conversely to the Horsehead and IC63, we are able to fit simultaneously the four IRAC bands, including the band at 4.5 µm. This is interesting as albeit the bands at 3.6, 5.8, and 8 microns cover aromatic and aliphatic features, the 4.5 band almost only covers the dust continuum (see Fig. 3 in Paper I). This means that contrary to the Horsehead and IC63 cases, the band to continuum ratio is well reproduced in the Orion Bar (as well as in protoplanetary disks, see Boutéraon et al. 2019 for instance). Therefore, THEMIS in highly irradiated regions (Orion Bar, protoplanetary disks, etc.) seems to work well, however, it needs some further developments in low to moderate PDRs. Spectral observations that will be accessible with the JWST will most likely bring more conclusive constraints.

6 Discussion

Here, we present the constraints on dust properties we obtain in IC63 and the Orion Bar (Sect. 6.1). We also summarise the results in the Horsehead from Paper I. Thence we present the mechanisms that are most likely at the origin of the nanograin evolution in PDRs, whether for their destruction (Sect. 6.2) or formation (Sect. 6.3). We finally discuss these results and propose a scenario for dust evolution in PDRs (Sect. 6.4).

6.1 Main results

We use the 3D radiative transfer code SOC together with THEMIS to model dust emission in IC63 and the Orion Bar. We find a good agreement between our modelled dust emission and the observations of Spitzer and Herschel in IC63 (see Fig. A.1) and the Orion Bar (see Fig. D.1).

We first constrain the depth threshold above which the density profile reaches a maximum (i.e. z₀) only based on the width of the dust modelled emission, which does not depend on the dust properties. We then assess dust properties in IC63 and in the Orion Bar. The main results are as follows:

1. Whether in IC63 or in the Orion Bar, the nano-grain dust-to-gas mass ratio, M_a-C/M_H, is lower than in the diffuse ISM. In IC63, M_a-C/M_H is roughly twice as low as in the diffuse ISM. In the Orion Bar, M_a–C/M_H is 60–100 times lower than in the diffuse ISM. The uncertainty on this last value is due to the degeneracy between a_min,a-C and α.

2. The nano-grain minimum size, a_min,a–C, in IC63 is 1.75 times larger than in the diffuse ISM. In the Orion Bar, the degeneracy between a_min,a-C and α does not allow us to draw a conclusion on the nano-grain minimum size.

3. The power-law exponent of the nano-grain size distribution, α, is the same in IC63 and in the diffuse ISM. Regarding the Orion Bar, α is at least 1.2 times lower than in the diffuse ISM.

In IC63, it is possible to simultaneously fit the observations in all the photometric bands only if we remove the one at 4.5 µm.

This problem has already been encountered in the Horsehead Nebula (see Paper I) and explained therein. Regarding the Orion Bar, we do not have this problem because, whether we use this band or not, we still have a degeneracy between a_min,a__C and α. In addition, this does not affect our results on the dust properties. These results confirm what was found for the Horsehead, namely, an increase in the nano-grain minimum size together with a decrease in the nano-grain dust-to-gas mass. For illustrative purposes, the spectra obtained for the Orion Bar are used in Berné et al. (2022) and compared to spectral observations at several representative PDR positions. We also show, in Appendix E, a study of the NIR-to-FIR emission ratio across the Orion bar that is purely observation-based and therefore not model-dependent, and which is in agreement with a decrease in the nano-grain abundance when one moves away from the illuminating star.

Fig. 7 χ2 minimisation for the Orion Bar. Left: ${\chi }_{\text{MIR}}^2$ in the 2D space (amin,a–C and α). Right: ${\chi }_{\text{tot}}^2$ as a function of Ma_c/MH for five different couple amin,a–C/α that are represented by the five symbols on the left figure. 1 (black dot): 0.35 nm/−5.96, 2 (blue triangle): 0.51 nm/−5.96. 3 (orange square): 0.63 nm/−7.44, 4 (green star): 0.71 nm/−8.92, 5 (pink diamond): 0.77 nm/−12.25. We also indicate with black arrows the Ma_c/MH obtained for the Horsehead (see Paper I), IC63 (see Sect. 5.2), and the diffuse ISM (Jones et al. 2013, 2017). The vertical grey stripe corresponds to the range of Ma_c/MH that minimise ${\chi }_{\text{tot}}^2$ in the Orion Bar.

Fig. 8 Comparison between the observed dust emission and the modelled dust emission using the best set of dust parameters, model 2 in that case (see Fig. 7) in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The dust observed emission is shown in green line. The cut considered across the Orion Bar is shown in Fig. A.1.

6.2 Photo-destruction of nano-grains

In addition to being stochastically heated, nano-grains can be destroyed by energetic photons (e.g. Alata et al. 2014, 2015). As a-C(:H) nano-grains are composed of a mix of different molecular domains, that is, aromatic domains connected by aliphatic (C-C) and olefmic (C=C) bridges, their photo-destruction is triggered by the photo-dissociation of aliphatic and olefmic bonds. From there, there are at least three processes that can lead to the photo-destruction of nano grains: direct dissociation, thermal (vibrational) dissociation, and Coulomb explosion (see Montillaud et al. 2013, and references therein).

However, the probability that an absorbed photon leads to one of these processes is hard to estimate and it is therefore a challenge to derive the photo-destruction timescales. It is nonetheless possible to understand, in a qualitative sense, how the photo-destruction timescale evolves from one PDR to another. If the frequency of photon absorption increases (i.e. the absorption timescale τ_abs decreases) and the energy of these absorbed photons increases as well, the nano-grain has more chances to be photo-destroyed. Finally, in PDRs where the energy of the absorbed photons is almost identical, the ratio of the photo-destruction timescales can be approximated by the ratio of the absorption timescales.

Prior to their photo-destruction, nano-grains must have time to be photo-processed⁹. Thus, we need to compare the absorption timescale to the advection timescale, τ_ad, that is, the time that the incident UV light needs to heat up and dissociate the molecular gas at the cloud border. In practice, τ_abs(a) must be smaller than τ_ad to allow nano-grains to be photo-processed. The advection timescale is defined as τ_ad = L/v_DF where L is the typical scale over which the radiation field penetrates the PDR which we assume corresponds to the position of the maximum emission in the NIR (i.e. where nano-grains are emitting). Following Goldshmidt & Sternberg (1995), the velocity of the dissociation front v_DF can be defined as:

$$v_{\text{DF}}=71\times \left(\frac{G_0}{{10}^3}\right){\left(\frac{n_{\text{H}}}{{10}^4\text{H\hspace{0.17em}}{\text{cm}}^{-3}}\right)}^{-1}\text{km\hspace{0.17em}}{\text{s}}^{-1},$$(10)

where n_H is the gas density at the position of the maximum NIR emission (i.e. at a distance, L). We summarise the velocities and the advection timescales found in Table 4. The advection timescales fall within the range τ_ad ~ 10³−10⁴ yr.

The timescale τ_abs(a) between two photons absorption by a dust grain with a size a is defined as:

$${\tau }_{\text{abs}}\left(a\right)={\displaystyle {\int }_{v_{\min }}^{v_{\max }}\pi a^2{Q}_{\text{abs}}\left(a,v\right)\frac{I_v}{hv}}\text{d}v,$$(11)

where Q_abs(a, v) is the absorption efficiency at frequency v for a dust size a and I_v is the specific intensity of the incident radiation field.

Figure 9 shows the absorption timescale as a function of the dust size for the Horsehead, IC63, and the Orion Bar. Regardless of the PDR and the dust size, these timescales are smaller than the advection timescales. We therefore assume that the nano-grains have time to be photo-processed in those three PDRs.

To assess whether dust grains are likely to be photo-destroyed in the Horsehead, IC63, and in the Orion Bar, we estimate the average energy of a photon which is absorbed by a dust grain, 〈E_bs〉 (a) = 〈hv_abs〉 (a), as a function of the dust size and for the illumination conditions in those three PDRs (see Fig. 9). We observe that 〈E_abs,Orion〉 > 〈E_{abs,Horsehead}〉 > 〈E_abs,IC63〉. This is consistent with the star temperature being higher in the Orion Bar than in the Horsehead, which is itself higher than that in IC63 (T_star = 38 000, 35 000, 25 000 K, respectively). This suggests that nano-grains are most efficiently photo-destroyed in the Orion Bar, then slightly less so in the Horsehead, and, finally, the least efficiently in IC63. Those energies are almost the same for the Orion Bar and the Horsehead, which suggests that the photo-destruction efficiency slightly changes between those two PDRs compared to IC63. We also observe that τ_{abs,Horsehead} > τ_abs,IC63 > τ_abs,Orion which is consistent with the intensity of the incident radiation field larger in the Orion Bar than in IC63, itself higher than in the Horsehead (G₀ = 26 000, 1100, 100, respectively). This suggests that nano-grains are more frequently photo-destroyed in the Orion Bar, then less so in IC63, and, finally, the least so in the Horsehead. The photo-destruction timescale in the Orion Bar is therefore the lowest among those three PDRs. Also, as the average energy of absorbed photons is almost identical in the Orion Bar and in the Horsehead, the ratio of the photo-destruction timescales in those two PDRs can be approximated by the ratio of the absorption timescales, which is τ_{abs,Horsehead}/τ_abs,Orion ~ 100. The photo-destruction timescale must therefore be about two orders of magnitude larger in the Horsehead than in the Orion Bar.

Also, photo-destruction timescales are most likely proportional to the number of C-C and C=C bonds, itself proportional to the number of carbons in a dust grain. As this number increases with a³, there is probably a threshold in carbon number above which dust grains are difficult to destroy. In addition, if the photo-dissociation of C-C bonds does not happen frequently, carbonaceous grains might have time to accrete other carbon atoms from the gas phase or going through internal re-structuration that will make them more difficult to destroy. Nevertheless, these mechanisms are the same for the three PDRs and therefore our comparative study of the direct photo-destruction timescales in these PDRs is still legitimate.

Fig. 9 Absorption timescale (full line) and average energy of an absorbed photon (dashed-line) as a function of the dust size for the Horsehead (red), IC63 (blue), and the Orion Bar (green).

6.3 (Re-)Formation of nano-grains through the fragmentation of larger grains

Stars form in dense clouds, where dust grains are expected to be large aggregates (e.g. Köhler et al. 2015) and, therefore, where nano-grains are no longer existing. The radiative feedback of these freshly formed stars on their parent dense environments creates photon-dominated regions. The presence of nano-grains in these regions that were previously only filled with dust aggregates indicates that those nano-grains are most likely formed by the fragmentation of aggregates. Different processes can lead to the fragmentation of large grains into smaller grains, such as collisions, absorption of energetic photons, and Coulomb fragmentation. As these two last processes are likely to be less efficient because absorbed photon energy is rapidly dissipated in a grain (on the order of the vibrational timescale ~10⁻¹³ s) and because large grains will not be sufficiently charged in PDRs to undergo Coulomb fragmentation, we focus here on the fragmentation driven by grain collisions.

In the framework of THEMIS, when the local gas density increases, large grains can form a second mantle either through accretion of C and H atoms, available in the gas phase or through coagulation of a-C nano-grains on the surfaces of larger grains. These grains are called core-mantle-mantle (CMM) grains. In denser regions, CMM grains coagulate together to form aggregates (Köhler et al. 2015) called aggregate-mantle-mantle (AMM) grains. The size distribution of AMM can be found in Fig. 3 of Paper I. From now on, we use the AMM from THEMIS to model aggregates.

Using a classical one-dimensional approach, we estimate the drift velocities of aggregates driven by the radiative pressure caused by a blackbody with a temperature T_*, B_λ(T_*), as well as the collision timescales associated. We consider an homogeneous medium filled with aggregates and gas, where those that are close to the star are pushed away. The gravitational force of the star on the dust grain can be non-negligible in some cases, for instance, in AGB environments (e.g. Woitke 2006) and in circumstellar disks (e.g. Vinković 2009; Arnold et al. 2019). We therefore consider the gravitational force in our model. In this depiction (see Fig. 10), an aggregate is subject to three forces:

1. The radiative pressure force, F_pr(a), which is oriented along the star-aggregate axis;

2. the force due to the drag caused by collisions with the gas, F_drag(a), oriented in the opposite direction to the radiative pressure force;

3. the graviational force, F_grav(a), exerted by the illuminating star on the aggregate.

The equation of motion of a dust aggregate with a mass, m_dust, and a size, a, is written as follows:

$$m_{\text{dust}}\frac{d{v}_{\text{dust}}\left(a\right)}{\text{d}t}=F_{\text{pr}}\left(a\right)-F_{\text{drag}}\left(a\right)-F_{\text{grav}}\left(a\right).$$(12)

The corresponding forces are defined as:

$$F_{\text{drag}}\left(a\right)=\pi a^2{\rho }_{\text{gas}}{v}_{\text{dust}}\left(a\right)\sqrt{v_{\text{dust}}^2\left(a\right)+\frac{128k_B{T}_{\text{gas}}}{9\pi {\mu }_{\text{H}}}},$$(13)

$$F_{\text{grav}}\left(a\right)=\frac{4}{3}\pi a^3{\rho }_{\text{dust}}\times \left(\frac{G{M}_{*}}{D^2}\right),$$(14)

$$F_{\text{pr}}\left(a\right)={\displaystyle {\int }_{\lambda }{F}_{\text{pr}}\left(a,\lambda \right)\text{d}\lambda },$$(15)

where

$$F_{\text{pr}}\left(a,\lambda \right)=\pi a^2{Q}_{\text{pr}}\left(a,\lambda \right)\left(\frac{\pi R_{*}^2}{D^2}\times \frac{B_{\lambda }\left(T_{*}\right)}{c}\right).$$(16)

In the equations above, the drag force F_drag is taken from Jones et al. (1996) and the radiative pressure force, F_pr, from Abergel et al. (2002). In this force, the term in $v_{\text{dust}}^2$ under the square root is associated with the intrinsic velocity of the dust grain while the second term corresponds to the gas particle velocity due to brownian motions. The different parameters are: ρ_gas = µ_H n_H as the mean gas density, µ_H = 1.4 m_H as the mean atomic mass of the gas, m_H as the H atom mass, T_gas as the gas temperature, ρ_dust as the dust density, and D is the distance from the aggregate to the star, while nH is the gas density. The radiation pressure efficiency is defined as:

$$Q_{\text{pr}}=Q_{\text{abs}}+Q_{\text{sca}}\times \left(1-g\right),$$(17)

where Q_abs (Q_sca) is the dust absorption (scattering) efficiency and g is the anisotropy factor of the scattering phase function.

In our case, the velocities of interest lead us to consider that the brownian motion of gas particles is, to first order, negligible compared to the dust motion hence Eq. (13) can be simplified to:

$$F_{\text{drag}}\left(a\right)=\pi a^2{\rho }_{\text{gas}}{v}_{\text{dust}}^2\left(a\right).$$(18)

Using Eqs. (14), (15), and (18) in Eq. (12), we obtain a first-order equation whose asymptotic solution corresponds to the drift velocity:

$$v_{\text{drift}}\left(a\right)=\frac{1}{D}{\left[\frac{1}{1.4m_{\text{H}}{n}_{\text{H}}}\left(\frac{\pi R_{*}^2}{c}\langle Q_{\text{pr}}{B}_{\lambda }\rangle -\frac{4}{3}a{\rho }_{\text{dust}}G{M}_{*}\right)\right]}^{1/2},$$(19)

with

$$\langle Q_{\text{pr}}{B}_{\lambda }\rangle ={\displaystyle {\int }_{\lambda }{Q}_{\text{pr}}\left(a,\lambda \right)B_{\lambda }}\left(T_{*}\right)\text{d}\lambda .$$(20)

As we are interested in the regions of PDRs where nanograins exist (i.e. where NIR and MIR dust emission reaches a maximum), we calculate the drift velocity for two extreme values of the gas density that are n_H = 10⁴ and 10⁵ Hcm⁻³, which define the slice where fragmentation takes place.

We show, in Fig. 11, the drift velocities of AMM in IC63, the Horsehead, and the Orion Bar for n_H varying from 10⁴ to also stick together and lead to grain growth. Guttler et al. (2010) found that when two aggregates with similar sizes collide with a velocity larger than v_frag = 1m s⁻¹, they will both be fragmented into smaller species according to a power-law size distribution n(a) ∝ a^γ with γ that varies from −2.1 to −1.2. As regardless of the size and the PDR, the drift velocity is at least 10⁴ larger than v_frag (see Fig. 11, top panel), we can assume that all collisions between AMM in IC63, the Horsehead, and the Orion Bar, lead to their fragmentation. It is also important to note that γ varies from −2.1 to −1.2, which means that aggregates are most likely fragmented into many very small grains rather than a few small grains.

From there, it is possible to estimate the fragmentation timescale from the collision timescale τ_coll(a₁, a₂) (i.e. the collision time between two aggregates of sizes a₁ and a₂) defined as follows:

$${\tau }_{\text{coll}}\left(a_1,a_2\right)={\left(n_{\text{H}}\sqrt{n_{\text{d}}\left(a_1\right)n_{\text{d}}\left(a_2\right)}\pi {\left(a_1+a_2\right)}^2{v}_{\text{rel}}\left(a_1,a{}_2\right)\right)}^{-1},$$(21)

where n_d(a) is the relative abundance of AMM grains with size a and v_rel(a₁, a₂) the relative velocity between two AMM. As the two AMM grains are pushed inwards the same direction and because the velocity of a grain located deeper in the PDR is much lower¹⁰ than the grain closer to the star, the relative velocity between the two grains is: v_rel(a₁, a₂) ≃ v_drift(a₁).

We show in Fig. 11 (bottom panel) the collision timescales between an AMM grain with size a and an AMM grain with size a₀ ~ 0.05 µm, the most abundant AMM. As the collision timescales are lower than the advection timescales, the fragmentation of large grains is a viable scenario for the formation of nano-grains in PDRs.

Fig. 10 Schematic view of the nano-grain formation through fragmentation driven by collisions of larger grains, followed by the nano-grain photo-destruction.

Fig. 11 Drift velocities and collision timescales. Top: drift velocities for IC63 (blue), Horsehead (red), and the Orion Bar (green) with nH varying from 104 to 105 H cm−3. Bottom: collision timescales between an AMM grain with size a colliding with another AMM with size a0 ∼ 0.05 μm.

6.4 Dust evolution scenario

The fragmentation of larger grains due to collisions caused by radiative pressure appears to be an efficient mechanism to form nano-grains. The efficiency of this mechanism mostly depends on G₀ whereas barely depends on the temperature of the illuminating star. We therefore expect the nano-grain formation to be more efficient in IC63 than in the Horsehead, and then more efficient in the Orion Bar than in IC63. On the other hand, the nano-grain photo-destruction efficiency depends on both the star temperature and G₀. Conversely to the Orion Bar and the Horsehead, the average energy of the incident photons in IC63 is not large enough to efficiently photo-destroy nano-grains. We explain this scenario in detail here.

6.4.1 Dependence of the nano-grain formation and destruction efficiencies with PDR depth

Because of dust extinction, the UV radiation field decreases with depth inside the PDR, which affects the radiative pressure and therefore the nano-grain formation. Also, as the nano-grain destruction depends on the amount of photons, the destruction of nano-grains is strongly affected by dust extinction. In order to easily compare one PDR to another, we explain the following in terms of A_V which additionally takes the density profile into account.

We show, in Fig. 12, the drift velocity for the three PDRs as a function of A_V We see that regardless of the PDR, the velocity drift is greater than v_lim ~ 1m s⁻¹ (velocity above which collisions between dust grains lead to fragmentation) until A_V ~ 13.5 (A_V,lim ~ 13.5,18.5, and 19.5 for the Horsehead, IC63, and the Orion Bar respectively). The net decrease in for large values of A_V is due to the fact that the radiative pressure force is lower than the gravitational force above these values of A_V hence the grain is attracted to the star (i.e. negative velocity drifts, not showed in this figure). This explains why we find aggregates in denser parts of PDRs and also why they are not all affected by fragmentation due to collisions.

We show, in Fig. 13, the collision timescales between the most abundant aggregates (AMM with a₀ ~ 0.05 µm) for the three PDRs. If these timescales are larger than the advection timescales, collisions should not occur. This provides us with an upper limit on A_V above which formation of nano-grains through the fragmentation is no longer efficient. These limits are A_V,lim~3.1−3.9, 2.1−4.2, and 2.1−3.7 for IC63, Horsehead, and the Orion Bar, respectively. In these ranges, the drift velocity is larger than the velocity threshold, hence all collisions lead to fragmentation (see Fig. 12). Also, it means that above these values, aggregates must exist − which is consistent with the existence of aggregates above A_V > 2–3 (Ysard et al. 2013, 2016). Most importantly, the nano-grain maximum emission occurs for A_V < 0.5 and because our scenario of nano-grain formation through fragmentation of aggregates is efficient below A_V ~ 2.1–3.9 (depending on the PDR), this reinforces our scenario of nano-grain formation.

We also show, in Fig. 13, the absorption timescales. In order to be photo-destroyed, nano-grains must have time to absorb photons. In other words, τ_abs must be lower than the advection timescale. Based on this, we obtain limits in A_V above which nano-grains are unlikely to be photo-destroyed. We note that this upper limit is therefore set at lower values of A_V as only a small fraction of the absorbed photons leads to the photo-destruction of nano-grains. Nevertheless, these limits are A_V,lim ~ 2.1–2.2, 1.85–2.05, and 2.1–2.2 for IC63, Horsehead, and the Orion Bar, respectively. As nano-grain maximum emission happens for A_V < 0.5 and because nano-grains can be photo-processed below A_V ~ 1.85–2.2 (depending on the PDR), they are most likely subject to photo-destruction below this threshold in A_V. The mechanisms at the origin of the nano-grain formation and destruction are thus expected to be efficient where nano-grains emit in the irradiated outer part of PDRs.

Fig. 12 Drift velocity as a function of Av for IC63 (blue line), Horsehead (Red line), and Orion Bar (green line). The horizontal black line corresponds to the velocity threshold above which collisions lead to fragmentation (see Sect. 6.3). The vertical dashed lines correspond to the limit in Av above which the velocity drift is not high enough to generate fragmentation.

6.4.2 Evolution of the nano-grain abundance

As nano-grains are efficiently formed in IC63 but not efficiently photo-destroyed, compared to the Horsehead, they are expected to be more abundant in IC63 than in the Horsehead. Regarding the Orion Bar, the discussion is slightly more challenging. Compared to the Horsehead, nano-grains are more efficiently photo-destroyed but also more efficiently formed through the fragmentation of larger grains. To first order, we assume that the efficiency of the nano-grain formation is proportional to the collision timescale and that the efficiency of the nano-grain destruction is proportional to the absorption timescale. We note that regardless of the dust size, the collision timescale is a factor ~10 larger in the Horsehead than in the Orion Bar (see Fig. 11, bottom panel), whereas the absorption timescale is a factor ~100 greater in the Horsehead than in the Orion Bar (see Fig. 9). This means that the ratio between the destruction efficiency and the formation efficiency is ~ 10 times greater in the Orion Bar than in the Horsehead. This supports our results as we find that the nano-grain abundance is lower by a factor of at least 10 in the Orion Bar compared to the Horsehead, which suggests that the ratio between the mechanism that leads to the nano-grain destruction with the one leading to their formation shown to be greater in the Orion Bar than in the Horsehead.

6.4.3 Evolution of the nano-grain minimum size and power-law size-distribution exponent

In the diffuse ISM, the nano-grain minimum size is around 0.4 nm, (Jones et al. 2013) whereas it increases in PDRs to ~0.7 nm in IC63, ~0.8 nm in the Horsehead (see Paper I), and up to a maximum of ~0.8 nm in the Orion Bar. This increase of 0.1 nm in the nano-grain minimum size from IC63 to the Orion Bar and Horsehead implies an noticeable increase of 1.5 in the carbon atom number, which is expected as the average energy of absorbed photons in both the Orion Bar and Horsehead is larger than in IC63. Also, as the nano-grain minimum size varies little from one PDR to another, the photo-destruction timescale must drastically decrease above a given size limit around 0.7–0.8 nm. Allain et al. (1996) showed that despite the fact that the photo-destruction timescale barely varies from one small PAH to another (i.e. N_C < 30), it can vary broadly from a small PAH to a larger one (N_C ~ 50). Indeed, photo-destruction timescales are expected to be about 10² to 10³ years for PAHs going from benzene (C₆H₆) to ovalene (C₃₂H₁₄) while it can reach ~10¹⁰ years for a PAH with 50 carbon atoms. As the photo-destruction of nano-grains depends on the number of C-C/C=C bonds and because this number evolves with a³, one can understand that the photo-destruction timescales evolves quickly with N_C (N_C ∝ α³).

In the diffuse ISM, the exponent of the THEMIS nano-grain power-law size distribution is α = −5. In the Horsehead, we find a lower value α ~ − 6 and α < −6 in the Orion Bar. This indicates that the fragmentation of large grains into nano-grains favours the formation of many small nano-grains instead of few larger nano-grains. The fact that we find the same value of α in IC63 as in the diffuse ISM supports the scenario where the formation efficiency of nano-grains through the fragmentation is larger than the photo-destruction efficiency. Indeed, this suggests that the nano-grains population had nearly enough time to completely reform in IC63, unlike the Horsehead and Orion Bar.

Fig. 13 Absorption timescales are represented by the green lines for IC63 (left panel), Horsehead (middle panel), Orion Bar (right panel). Collision timescales between the most abundant aggregates (a0 ~ 0.05 µm) are shown in red lines. The horizontal grey stripes correspond to the advection timescales (see Table 4). The vertical light green (red) stripes correspond to the region where τabs ~ τad (τcoll ~ τad).

7 Conclusion

We use Spitzer and Herschel data to map dust emission in two PDRs: IC63 and the Orion Bar. We modelled dust emission across those two PDRs using the THEMIS dust model together with the radiative transfer code SOC. We show that it is possible to partially constrain the gas density profile only based on the width of the dust emission profile. Using these density profiles, we show that dust similar to that of the diffuse ISM cannot explain the observations and, thus, the dust size distribution has to be modified.

The nano-grain dust-to-gas mass ratio is roughly half that of IC63 and almost 100 times lower in the Orion Bar than in the diffuse ISM. The nano-grain minimum size is about 0.7 nm in IC63 and no more than 0.8 nm in the Orion Bar. The slope of the size distribution in IC63 is almost the same as in the diffuse ISM whereas it is steeper in the Orion Bar. This suggests that the mechanism at the origin of the nano-grain destruction is more efficient than the one of nano-grain formation in the Orion Bar compared to IC63.

To this end, we estimate formation timescales assuming that nano-grains are mainly formed through collisions between larger grains driven by radiative pressure from the star. We also estimate destruction timescales assuming that nano-grains are destroyed by energetic photons. Based on this timescale analysis, we find that the nano-grain destruction-to-formation ratio increases from IC63 to the Orion Bar, through the Horsehead, which explains the decrease in the nano-grain abundance from IC63 to the Orion Bar. As the photo-destruction efficiency is quite low in IC63 compared to the Orion Bar and Horsehead, whereas the formation of nano-grains in IC63 is efficient, this explains why the presence of dust in IC63 is more similar to what we would find in the diffuse ISM – as compared to what we would find in the Orion Bar and the Horsehead. The fact that the nano-grain minimum size scarcely varies from one PDR to another, but is still twice as large as than in the diffuse ISM suggests that there is a critical size above which dust grains are resilient to photo-destruction, regardless of irradiation. Based on our constraints on the nano-grain minimum size, we find that this critical size is around 0.7–0.8 nm.

Acknowledgements

We thank the anonymous referee for very helpful suggestions and comments. This work was supported by the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. We would like to thanks Olivier Berné for stimulating discussions on dust photo-dissociation. T.S. acknowledges support from the Knut and Alice Wallenberg Foundation (grant no. 2020.0081).

Appendix A IC63 and the Orion Bar as seen with Spitzer/IRAC and Herschel/PACS

Fig. A.1 IC63 and the Orion Bar as seen with Spitzer and Herschel. Top six figures: IC63 seen in six photometric bands (3.6, 4.5, 5.8, 8, 24, and 70 µm). Bottom five figures: the Orion Bar seen in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The white solid lines correspond to the cuts used in our study.

Appendix B Influence of variations in dust properties on the width of the emission profiles

Figure B.1 shows the FWHM of the dust emission profile at 3.6 µm and 70 µm in the 2D space defined by M_a–C/M_H and a_{min, a–C}^.

Fig. B.1 FWMH in the 2D space (Ma–C/MH, amin,a–C). Top: Full width at half maximum (FWHM) of the dust emission profile at 3.6 µm in the 2D space (Ma–C/MH, amin,a–C). Bottom: Same at 70 µm.

Appendix C minimisation in the 3D space (M_a__C/M_H, a_min,a–C, and α), IC63

Fig. C.1 ${\chi }_{\text{tot}}^2$ in the 2D space (Ma–C/MH, amin,a–C). Each case corresponds to a different value of α. The black dashed lines locate the minimum value of ${\chi }_{\text{tot}}^2$.

Appendix D Comparison between the dust modelled and observed emission in the Orion Bar

Fig. D.1 Comparison between the observed dust emission and the modelled dust emission using the best set of dust parameters (from model 1 on the top to model 5 on bottom, showed in Fig. 7) in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The dust observed emission is shown in green lines. The cut considered across the Orion Bar is shown in Fig. A.1.

Appendix E NIR to FIR ratio across the Orion bar

Here, we study the nano-grain to large grain abundance ratio across the Orion Bar via the NIR-to-FIR emission ratio. This approach purely observational-based allows us to qualitatively follow dust evolution without being model-dependent. We show in Fig. E.1 the normalised intensities in nine photometric bands (top panel) and the NIR-to-FIR ratio (bottom panel) across a selected cut in the Orion Bar (this cut is shown in Fig. 1).

We face a challenge here as the spatial resolution of the NIR observations with Spitzer is much better than that of the RIR observations with Herschel. In fact, while we have many points available within the PDR with Spitzer, we only have around three points from the HII regions via the PDR to the dense region. It is therefore challenging to conclude on the nano-grain to large grain abundance ratio as a function of depth inside the PDR. Despite this difficulty, we do, however, see a net trend as this ratio decreases in the PDR region when we move away from the illuminating star, which is in agreement with a decrease in the nano-grain abundance with depth inside the PDR.

This study is limited by the fact that the spatial resolution of the NIR emission is much better than the one of the RIR emission. However, to end on a positive note, it is thanks to JWST that there is indeed a way to follow nano-grains and large grains with the same spatial resolution: large grains emit in the FIR but scatter light in the visible/NIR. Therefore, we normally would be able to follow large grains through scattering in the visible/NIR with almost the same spatial resolution than the nano-grains through their NIR/MIR emission. This would hopefully provide us with an accurate nano-grain to large grain abundance ratio as a function of depth inside PDRs in the near future.

Fig. E.1 Top: Normalised emission across a selected cut in the Orion Bar (This cut is shown in Fig. 1) in nine photometric bands. Botton: NIR to FIR emission ratio across the Orion Bar. The three different zones correspond to the molecular region, the PDR, and the HII regions respectively.

Appendix F Uncertainties on the advection timescale

The uncertainty on the advection timescale is defined as:

$$\frac{\text{Δ}{\tau }_{\text{ad}}}{{\tau }_{\text{ad}}}=\sqrt{{\left(\frac{\text{Δ}L}{L}\right)}^2+{\left(\frac{\text{Δ}{n}_{\text{H}}}{n_{\text{H}}}\right)}^2+{\left(\frac{\text{Δ}{G}_0}{G_0}\right)}^2},$$(F.1)

where the uncertainties on L, n_H, and G₀ are defined in Table 4.

References

All Tables

All Figures

	Fig. 1 Selected PDRs seen with Spitzer. Top: IC63 seen at 3.6 µm. Bottom: Orion Bar seen at 3.6 µm. The white solid lines correspond to the cuts used in our study.
In the text

	Fig. 2 Dust size distributions for a-C (solid line), a-C:H/a-C (dotted line), and a-Sil/a-C (dotted dashed line).
In the text

	Fig. 3 Comparison between the observed dust emission and the modelled dust emission in IC63 using the best set of dust parameters (Ma_C/MH = 0.10 × 10−2, amin.a-C = 0.70 nm, and α = −5) in six photometric bands (3.6, 4.5, 5.8, 8, 24, and 70 µm). The dust modelled (observed) emission is shown in blue (green) line. The cut considered across IC63 is shown in Fig. A.1.
In the text

Fig. 4 ${\chi }_{\text{wid}}^2$ minimisation for IC63 and the Orion Bar. Top: FWHM of the observed dust emission profiles in different photometric bands as a function of zo (left for IC63 and right for the Orion Bar). Bottom: ${\chi }_{\text{wid}}^2$ as a function of z0 (left for IC63 and right for the Orion Bar). The vertical black dashed lines corresponds to the minimum value of ${\chi }_{\text{wid}}^2$. Radiative transfer has been done using diffuse ISM-like dust with an incident radiation field corresponding to a blackbody at 25 000 K (38 000 K) with G0 = 1100 (G0 = 2.6 × 104) for IC63 (Orion Bar).

In the text

	Fig. 5 ${\chi }_{\text{tot}}^2$ in the 2D space (Ma_C/MH and amin, a-C) for α = −5 in the IC63 model.
In the text

	Fig. 6 Top: Ma-C/MH and amin,a-C associated with the minimum ${\chi }_{\text{tot}}^2$ in the 2D space (Ma-C/MH, amin,a-C) as a function of α (see Fig. C.1). Bottom: minimum value of ${\chi }_{\text{tot}}^2$ in the 2D space (Ma-C/MH, amin,a-C) as a function of α.
In the text

Fig. 7 χ2 minimisation for the Orion Bar. Left: ${\chi }_{\text{MIR}}^2$ in the 2D space (amin,a–C and α). Right: ${\chi }_{\text{tot}}^2$ as a function of Ma_c/MH for five different couple amin,a–C/α that are represented by the five symbols on the left figure. 1 (black dot): 0.35 nm/−5.96, 2 (blue triangle): 0.51 nm/−5.96. 3 (orange square): 0.63 nm/−7.44, 4 (green star): 0.71 nm/−8.92, 5 (pink diamond): 0.77 nm/−12.25. We also indicate with black arrows the Ma_c/MH obtained for the Horsehead (see Paper I), IC63 (see Sect. 5.2), and the diffuse ISM (Jones et al. 2013, 2017). The vertical grey stripe corresponds to the range of Ma_c/MH that minimise ${\chi }_{\text{tot}}^2$ in the Orion Bar.

In the text

	Fig. 8 Comparison between the observed dust emission and the modelled dust emission using the best set of dust parameters, model 2 in that case (see Fig. 7) in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The dust observed emission is shown in green line. The cut considered across the Orion Bar is shown in Fig. A.1.
In the text

	Fig. 9 Absorption timescale (full line) and average energy of an absorbed photon (dashed-line) as a function of the dust size for the Horsehead (red), IC63 (blue), and the Orion Bar (green).
In the text

	Fig. 10 Schematic view of the nano-grain formation through fragmentation driven by collisions of larger grains, followed by the nano-grain photo-destruction.
In the text

	Fig. 11 Drift velocities and collision timescales. Top: drift velocities for IC63 (blue), Horsehead (red), and the Orion Bar (green) with nH varying from 104 to 105 H cm−3. Bottom: collision timescales between an AMM grain with size a colliding with another AMM with size a0 ∼ 0.05 μm.
In the text

	Fig. 12 Drift velocity as a function of Av for IC63 (blue line), Horsehead (Red line), and Orion Bar (green line). The horizontal black line corresponds to the velocity threshold above which collisions lead to fragmentation (see Sect. 6.3). The vertical dashed lines correspond to the limit in Av above which the velocity drift is not high enough to generate fragmentation.
In the text

Fig. 13 Absorption timescales are represented by the green lines for IC63 (left panel), Horsehead (middle panel), Orion Bar (right panel). Collision timescales between the most abundant aggregates (a0 ~ 0.05 µm) are shown in red lines. The horizontal grey stripes correspond to the advection timescales (see Table 4). The vertical light green (red) stripes correspond to the region where τabs ~ τad (τcoll ~ τad).

In the text

	Fig. A.1 IC63 and the Orion Bar as seen with Spitzer and Herschel. Top six figures: IC63 seen in six photometric bands (3.6, 4.5, 5.8, 8, 24, and 70 µm). Bottom five figures: the Orion Bar seen in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The white solid lines correspond to the cuts used in our study.
In the text

	Fig. B.1 FWMH in the 2D space (Ma–C/MH, amin,a–C). Top: Full width at half maximum (FWHM) of the dust emission profile at 3.6 µm in the 2D space (Ma–C/MH, amin,a–C). Bottom: Same at 70 µm.
In the text

	Fig. C.1 ${\chi }_{\text{tot}}^2$ in the 2D space (Ma–C/MH, amin,a–C). Each case corresponds to a different value of α. The black dashed lines locate the minimum value of ${\chi }_{\text{tot}}^2$.
In the text

	Fig. D.1 Comparison between the observed dust emission and the modelled dust emission using the best set of dust parameters (from model 1 on the top to model 5 on bottom, showed in Fig. 7) in five photometric bands (3.6, 4.5, 5.8, 8, and 70 µm). The dust observed emission is shown in green lines. The cut considered across the Orion Bar is shown in Fig. A.1.
In the text

	Fig. E.1 Top: Normalised emission across a selected cut in the Orion Bar (This cut is shown in Fig. 1) in nine photometric bands. Botton: NIR to FIR emission ratio across the Orion Bar. The three different zones correspond to the molecular region, the PDR, and the HII regions respectively.
In the text