<figure><img src="image_315.png" /></figure>

### III.1.2.4 The Matryoshka Effect

The main limitation of dust studies lies in the near impossibility to constrain at the same time the microscopic dust properties and their sub-pixel macroscopic spatial distribution. All the approaches we have discussed in this chapter face this problem. It can be illustrated with what Galliano et al. (2018) called the *matryoshka effect*. This empirical effect originates in the impact of the spatial resolution of the observations on the constrained parameters.

**Demonstration on the LMC.** Fig. III.21 demonstrates the effect with the modeling of the dust mass in a strip covering one fourth of the LMC, by Galliano et al. (2011). The different images on top show the spatial resolution that is used. The first image is one single large pixel. The second one is divided in four pixels, and so on. The curve in the bottom panel of Fig. III.21 shows the dust mass in the strip derived by summing all the pixels, at each resolution. We see that this mass increases with spatial resolution, until reaching a plateau around ≈ 10 pc. For this particular region, the discrepancy with the global mass is about ≈ 50%. Galliano et al. (2011) interpreted this effect by noticing that the cold dust, which accounts for most of the mass and, at the same time, is the least luminous, is diluted into the warm dust emission when we sum all the regions together. With spatial resolution however, we can better separate the bright and cold regions. It is thus reasonable to assume that the most accurate estimate of the dust mass is the one obtained at the highest resolution. This assumption is confirmed by the fact that the length-scale at the growth curve plateau (≈ 10 pc) corresponds roughly to the mean free-path of a U-band photon at a density $n_H \approx 100 \text{ cm}^{-3}$ (Table III.1; the LMC has a half-Solar metallicity; Pagel, 2003). This is the typical density of the CNM and diffuse $H_2$ phase (cf. Table III.6). It is possible that, if we could increase the resolution, we would see another plateau around $n_H \approx 10^4 \text{ cm}^{-3}$ (≈ 0.1 pc), corresponding to dense $H_2$ clouds.

**Generalization.** This effect has been independently confirmed by Galametz et al. (2012), Roman-Duval et al. (2014) and Aniano et al. (2020), although it is less important if the maximum spatial resolution is not as high as ours.

<figure><img src="image_316.png" /></figure>

☞ The dust mass derived at high spatial resolution is always larger than its global estimate.

### III.1.3 Application to Nearby Galaxies

We now review the application of SED models to observations of nearby galaxies, aimed at constraining the grain properties in different environments. We illustrate the different aspects using our own projects and collaborations.

#### III.1.3.1 The Different Types of Galaxies

There is a wide diversity of galaxy types. Several systems of classification have been developed, through the years. In particular, the Hubble-de Vaucouleurs system, although outdated, is still used nowadays.

**The outdated galaxy morphological classification.** The most famous observational system of morphological classification is the *Hubble tuning fork* or *Hubble-de Vaucouleurs diagram*, represented in Fig. III.22. It was originally developed by Hubble (1936), and refined by de Vaucouleurs (1959). It is based on the morphological characteristics of galaxies in the visible range: presence and thickness of spiral arms, bars, rings, etc. There are essentially three classes of galaxies (left, center and right parts of Fig. III.22): (i) ellipticals, also called *Early-Type Galaxies* (ETG); (ii) spirals, also called *Late-Type Galaxies* (LTG); and (iii) irregulars and dwarf spheroidals. There are sub-categories with abstruse

<figure><img src="image_317.png" /></figure>
<figure><img src="image_318.png" /></figure>

<figure><img src="image_319.png" /><figcaption>FIGURE III.21 – Matryoshka effect demonstrated on the LMC. On top are nine SPIRE$_{500µm}$ images of a strip covering about one fourth of the LMC, observed by Meixner et al. (2010). These images are rebinned at different spatial resolutions (indicated on the top). The bottom panel shows the dust mass derived by modeling the SED of each pixel of these images and summing them to derive the total mass in the strip (Galliano et al., 2011). It is normalized by the mass obtained when modeling the global (i.e. one pixel) SED of the strip, $M_{\text{dust}}^{\text{global}}$. Licensed under CC BY-SA 4.0.</figcaption></figure>

notations (SAa, E2, etc.) that would be a waste of time to describe here. Overall, this is a mid-XXth-century empirical classification, based on stellar dynamics, that does not take into account the IR information (especially the SF activity), nor the radio properties (presence of an AGN). Recently, with the DustPedia collaboration (Davies et al., 2017), we explored the sensitivity of dust properties to galaxy types (e.g. Davies et al., 2019; Bianchi et al., 2018; Nersesian et al., 2019). We did not find any clear systematic differences between adjacent sub-categories in Fig. III.22, and we found a large scatter within each class. There are overall trends between the three main categories, because they are linked to the gas fraction, metallicity and stellar populations. We will discuss those throughout this manuscript. The terminology “late/early-type” was introduced to denote the evolution of galaxies along the sequence. We now know that the sequence is reversed: early-type galaxies are the oldest objects, and late-types, the youngest ones. In addition, the most numerous galaxies in the local Universe, dwarf galaxies are under-represented in this diagram. They are part of the irregulars, which is a category by default. Finally, this Hubble-de Vaucouleurs classification was based on the local Universe, while galaxies at high redshift can exhibit different morphologies, such as clumpy chains and tadpoles (e.g. Elmegreen, 2015). We have developed an alternate, non-parametric classification,

<figure><img src="image_320.png" /></figure>
<figure><img src="image_321.png" /></figure>

taking into account IR morphology (Baes et al., 2020). It emphasizes the clumpy nature of the dust distribution in local galaxies.

<figure><img src="image_322.png" /></figure>

Stellar morphology is not particularly relevant to ISM studies.

<figure><img src="image_323.png" /><figcaption>FIGURE III.22 – Hubble-de-Vaucouleurs galaxy morphology diagram. Credit: A. Ciccolella / M. De Leo, Wikipedia, licensed under CC BY-SA 3.0.</figcaption></figure>

**Galactic properties that matter to ISM studies.** A few global quantities, such as the metallicity, the SFR or the gas fraction are more relevant to assess the general properties of the ISM of a galaxy. The three main categories of the Hubble-de Vaucouleurs diagram can be characterized the following way.

**Irregular galaxies** (e.g. Fig. III.23.a) can contain large amounts of atomic gas that typically extend to twice their Holmberg radius<fnref n="5" /> (e.g. Huchtmeier et al., 1981).

**At Solar metallicity,** irregulars are rich in dust. Their visible and MIR scale-lengths are very similar (Hunter et al., 2006). The dust emission in irregular galaxies is clumpy and warm, with hot dust and UIB emissions mostly observed towards bright HII regions. This suggests that massive stars are a major source of heating in these environments (e.g. Hunter et al., 2006).

**At low metallicity,** irregulars are dwarf galaxies. Dustiness increases with metallicity (cf. Sect. IV.3; Rémy-Ruyer et al., 2014; Galliano et al., 2021). The ISM in these objects is less dusty and thus, more transparent. Similarly to irregular galaxies, massive stars are a major source of heating in these objects (e.g. Walter et al., 2007), and they are permeated by SN-triggered shock waves (e.g. Oey, 1996; Izotov et al., 2007). Finally, these galaxies exhibit large HII envelopes. The IR emitting region can correspond to only ≈ 20–30% of the total mass of the system (e.g. Walter et al., 2007). A particularly important type of dwarf galaxies are *Blue Compact Dwarf Galaxies* (BCD). These galaxies are actively star forming. As their name indicates, they have blue optical colors, because they contain several SSCs, rich in young massive stars, and they are weakly attenuated by dust.

<figure><img src="image_324.png" /></figure>

<footnote n="5">5. The *Holmberg radius* is the radius within which the B-band surface brightness of the galaxy is 26.5 magnitudes per squared arcsecond.</footnote>
<figure><img src="image_325.png" /></figure>

**Late-type galaxies** (e.g. Fig. III.23.b) have a roughly Solar metallicity and an ISM accounting for ≈ 10–30% of their baryonic mass.

**Scale-lengths** of disk galaxies are of the order of a few kpc (up to ≈ 10 kpc). Their diameter tends to be systematically smaller in the MIR than in the visible (Malhotra et al., 1996), whereas the arm/inter-arm contrast is larger in the MIR than in the visible (Vogler et al., 2005). This is also seen in $^{12}$CO(J=1→0)$_{2.6mm}$, Hα$_{656.3nm}$ or radio continuum (Sauvage et al., 1996; Walsh et al., 2002; Vogler et al., 2005). This is because these different tracers are indicators of star formation activity, which is enhanced along the spiral arms. In the IR, the disk scale-length increases with wavelength (Hippelein et al., 2003), and is larger in the FIR than in the visible (Tuffs et al., 1996; Alton et al., 1998; Haas et al., 1998; Davies et al., 1999; Trewhella et al., 2000; Fritz et al., 2012; Casasola et al., 2017). This FIR colour gradient observed in the disk suggests that part of the FIR emission arises from grains heated by the radially decreasing diffuse ISRF, the extended parts of the disk being cold (Block et al., 1994; Stevens et al., 2005; Hinz et al., 2012). FIR scale-lengths do not vary strongly as a function of galaxy type and are on average ≈ 10% larger than the stellar scale-lengths (e.g. Muñoz-Mateos et al., 2009; Hunt et al., 2015).

**Scale heights** of disk galaxies are typically of the order of a few hundred parsecs. Outside our Milky Way, edge-on galaxies are ideal objects to constrain this parameter (e.g. Xilouris et al., 1999). Radiative transfer codes are robust tools to derive such structural parameters (cf. Sect. III.1.3.2).

**Early-type galaxies** (e.g. Fig. III.23.c) possess very little dust: the average dust-to-stellar mass ratio is ≈ 50 times lower than that of spiral galaxies (Smith et al., 2012; Galliano et al., 2021). Dust lanes are, however, commonly detected in elliptical galaxies (e.g. Sadler & Gerhard, 1985; Gomez et al., 2010). Jura et al. (1987) for instance found that half of nearby ellipticals are detected at IRAS wavelengths. Smith et al. (2012) found that elliptical galaxies detected at 250 µm tend to have higher X-ray luminosities. This was further explored by G21. We will come back to this point in Sect. IV.2.2.2.

### III.1.3.2 Large-Scale Dust Distribution in Galaxies

The dust spatial distribution, that is the value of various dust parameters in different regions or pixels of a galaxy, can be determined in the MW and nearby galaxies. This determination however requires good quality, homogenized multi-wavelength images of the studied objects. This is therefore significantly more complex than modeling the SED of a point source.

**Homogeneous multi-wavelength data sets.** To accurately model SEDs of galaxies, the observed fluxes must originate, at each wavelength, in the same region, and must trace only the emission we are modeling (i.e. the dust emission, the escaping stellar emission, etc.). The following artifacts can be encountered.

**Contaminations** in the telescope beam can have several origins. Most of them are displayed in Fig. III.24.

**Foreground zodiacal emission** originates in dust grains from the Solar system (e.g. Fixsen & Dwek, 2002; Reach et al., 2003; Rowan-Robinson & May, 2013; Krüger et al., 2019). Its intensity depends greatly on the angle above the planetary disk. It is prominent in the MIR (cf. Fig. III.24). The zodiacal emission is quite homogeneous at the typical angular scale of nearby galaxies (a few degrees or less). It can thus be efficiently subtracted, using the emission outside the target. Otherwise, several models compute its synthetic spectrum (e.g. Krüger et al., 2019).

<figure><img src="image_326.png" /></figure>
<figure><img src="image_327.png" /><figcaption>(a) Irregular, dwarf</figcaption></figure>

<figure><img src="image_328.png" /></figure>

(I Zw 18)

Low metallicity (≈ 1/35 Z☉)

<figure><img src="image_329.png" /><figcaption>(b) Spiral</figcaption></figure>

(M 33)

Solar metallicity (≈ 1/2 Z☉)

<figure><img src="image_330.png" /><figcaption>(c) Elliptical</figcaption></figure>

(Centaurus A)

High metallicity (≈ 3 Z☉)

**FIGURE III.23** – *Visible-range image of three nearby galaxies.* Panel (a) shows an image of the extremely low-metallicity, dwarf galaxy, I Zw 18, obtained with the instrument ACS onboard the HST (Aloisi et al., 2007). Panel (b) shows the spiral galaxy, M 33. Star-forming regions, traced by the Hα$_{656.3nm}$ line, are shown in red. Panel (c) shows the elliptical galaxy, Centaurus A. It has a warped dust lane and an AGN. Credit: (a) Cignoni & Tosi (2009), licensed under CC BY-SA 3.0; (b) Image Data: Subaru Telescope (NAOJ), Hubble Space Telescope – Image Processing: Robert Gendler – Additional Data: BYU, Robert Gendler, Johannes Schedler, Adam Block – Copyright: Robert Gendler, Subaru Telescope, NAOJ, with permission from Robert GENDLER; (c) courtesy of ESO, licensed under CC BY 4.0.

**Foreground Galactic emission** originates in dust grains from cirrus clouds within the MW. Its spectrum does not vary significantly with pointing direction, as the Galactic ISRF is relatively homogeneous. However, its total intensity scales with the column density of the ISM between the observer and the target galaxy. The H column density is the lowest at high Galactic latitude ($N_H \approx 10^{24}$ H/m$^2$). The fractal structure of interstellar clouds (e.g. Elmegreen & Falgarone, 1996) results in a high degree of structure at small angular scales. In other words, it is difficult to subtract this emission using off-target areas. This contamination can not always be subtracted. Surface brightness being independent of distance, the emission from the diffuse ISM of nearby galaxies is similar to that of the MW, making the study of the former particularly challenging.

**Background galaxy emission** constitutes the CIB (e.g. Dole et al., 2006). Its SED looks similar to the diffuse Galactic ISM (cf. Fig. III.24). Spatially, it is granular, as it is the sum of numerous point sources. It is difficult to accurately subtract. It is thus another component that makes studying the diffuse ISM of nearby galaxies challenging.

**Cosmological background emission** is a T = 2.73 K black body (cf. Sect. I.2.4.2; Mather et al., 1994). It contaminates essentially the mm regime. Its emission is globally isotropic with some fluctuations. The amplitude of these fluctuations is shown as a solid red line in Fig. III.24. These fluctuations are a bit more difficult to subtract. However, in general this source of contamination is not the most challenging to remove.

**Non-dusty contamination**, such as gas-phase line emission or the radio continuum need to be subtracted. For instance, in the submm regime, the CO lines and the free-free continuum can account for 10–20% of the emission around $\lambda \approx 1$ mm (cf. Fig. II.10 and e.g. Galliano et al., 2003, 2005). Additional independent constraints are necessary, such as spectroscopic observations of the contaminating lines, and long-wavelength radio continuum fluxes that probe the free-free and synchrotron emission.

**Differences in angular resolution** are due to variations of the beam size across the observed SED. If

<figure><img src="image_331.png" /></figure>
<figure><img src="image_332.png" /></figure>

not corrected, this can have dramatic consequences, especially if the studied region has large gradients of emission on scales of the order of the largest beam size. It is however easy to correct. One needs to identify, among the instruments used, which one has the largest beam, $\Omega_{max}$. One then simply needs to degrade the angular resolution of all the other wavelengths, to the resolution $\Omega_{max}$. This degradation is performed by convolving the images by a kernel, which is the $\Omega_{max}$ beam deconvolved by the beam at the nominal wavelength. Such kernels are provided, for instance, by Aniano et al. (2011) for a wide variety of instruments.

**Differences in field of view** come from the fact that the different instruments do not necessarily have the same orientation on the sky and the same pixel size. To model a spatially-resolved SED, it is thus necessary to reproject every image on a single grid. Numerous methods are available to perform this reprojection (e.g. Bertin et al., 2002). It can become problematic only when there are missing areas, such as incomplete maps or masked regions (e.g. because of saturation). The linear pixel size of the final grid can be as low as ≈ 1/3 of the largest beam size (Nyquist sampling).

**Uncertainties** must be taken into account as rigorously as possible. Ideally, we should not only determine the uncertainties of each pixel at each wavelength, but also their correlations. This can be simplified by separating the two major sources of uncertainties: noise and calibration effects. The noise comes from the instrument. It is usually provided by the data reduction pipeline. It thus needs to be propagated through the various sources of homogenization we have discussed. In particular, resolution degradation increases the median signal-to-noise ratio, as it has an averaging effect on the noise spatial distribution. It also creates correlations between pixels. To our knowledge, the best method to propagate noise uncertainties, which is also the simplest to implement, is by way of Monte-Carlo simulations (e.g. Galliano et al., 2011). The principle is the following.

1. We generate a large number, $N \approx 100$, of images with flux:

$$F_{\nu}^{(i)}(x, y, \lambda) = F_{\nu}(x, y, \lambda) + \delta(i, x, y, \lambda) \times \sigma_{\nu}(x, y, \lambda) \quad (III.41)$$

(i = 1, N), where $F_{\nu}$ and $\sigma_{\nu}$ are the flux and uncertainty coming from the data reduction pipeline. The random variable, $\delta$, is independent, normal with mean 0 and standard deviation 1.

2. We then perform the different homogenization steps (contamination subtraction, degradation, reprojection) on each random samples.

3. We now have, for each pixel and each wavelength of the final homogenized maps, a distribution of N values. The standard deviation of this distribution gives the uncertainty and we can compute correlation coefficients between different pixels or wavelengths.

Calibration uncertainties can be computed afterwards, as they are proportional to the flux. These uncertainties are fully correlated between pixels and partially correlated between wavelengths (e.g. G21).

This technical data preparation can be tedious, but it is crucial as dust model results directly rely on it. A significant effort has been put into providing homogenized databases of nearby galaxies. Among them, the most important surveys are the following.

* SAGE/HERITAGE (Meixner et al., 2006, 2013) were a series of surveys of the Magellanic clouds with *Spitzer* and *Herschel*.

* The DGS (Madden et al., 2013; Rémy-Ruyer et al., 2013) was a survey of nearby dwarf galaxies with *Herschel*, for which we also added *Spitzer* and ancillary data (Bendo et al., 2012; Rémy-Ruyer et al., 2015).

* SINGS/KINGFISH (Kennicutt et al., 2003, 2011) was a series of *Spitzer* and *Herschel* surveys of nearby galaxies.

* DustPedia (Davies et al., 2017; Clark et al., 2018) was a European collaboration project, which built a homogenized sample of all available data for ≈ 800 nearby galaxies.

<figure><img src="image_333.png" /></figure>
<figure><img src="image_334.png" /></figure>

<figure><img src="image_335.png" /><figcaption>FIGURE III.24 – *Principal sources of contaminations encountered when modeling SEDs.* The zodiacal spectrum (blue) has been computed using the model of Fixsen & Dwek (2002). The CMB spectrum is a $T = 2.73$ K black body (red dashed line). The solid red line corresponds to the level of CMB fluctuations. The CIB (green) comes from the compilation of Dole et al. (2006). The Galactic cirrus emission is the THEMIS model scaled by a column density $N_H = 10^{24}$ H/m$^2$. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Properties of individual galaxies.** Numerous studies have presented the SED modeling of nearby galaxies, and their derived dust properties, either globally or spatially-resolved. We have participated to several such projects (e.g. Whaley et al., 2009; Galametz et al., 2009; Boselli et al., 2010; Galametz et al., 2010; O'Halloran et al., 2010; Eales et al., 2010; Cortese et al., 2010; Sauvage et al., 2010; Bendo et al., 2010; Roussel et al., 2010; Gordon et al., 2010; Davies et al., 2010; Boquien et al., 2010; Skibba et al., 2012; De Looze et al., 2012a; Galametz et al., 2013; Ciesla et al., 2014; Gordon et al., 2014; Galametz et al., 2016; Bianchi et al., 2018; Nersesian et al., 2019). Reviewing them would be unwieldy, here. In general, these studies provide dust parameters (mass, starlight intensity, PAH mass fraction, etc.) of different objects, which can be used in combination with other tracers to refine our understanding of the studied source. They also provide scaling relations and calibrations of various diagnostics such as SFR tracers. These results can also be used to train machine-learning models that could predict the SED of a poorly-observed galaxy (e.g. Dobbels et al., 2020).

**Identifying dust heating sources.** A particular question, that has been tackled by several studies, is the identification of the sources responsible for dust heating within galaxies. In the MW, 3D reconstruction of the ISM distribution showed that the heating by young, O/B stars (Table III.2) was prominent in molecular regions, whereas the atomic phase was mainly heated by lower-mass stars (e.g. Sodroski et al., 1997; Paladini et al., 2007). In nearby galaxies, this depends on the SF activity of the galaxy. For instance, we showed that PAHs were essentially heated by field stars in the quiescent galaxy NGC 2403. These molecules are however heated by the escaping radiation from H II regions in the more actively star-forming object, M 83 (Jones et al., 2015). More generally, with the DustPedia sample, we found that dust in ETGs was mainly heated by old stars (Nersesian et al., 2019). It is only when considering more gas-rich galaxies that the contribution of young stars becomes more important. It can account for up to ≈ 60% of the dust luminosity in extreme late-type galaxies (Sm-Irr, Fig. III.22; Nersesian et al., 2019). These different heating sources have an impact on the global escape fraction (i.e. the fraction of stellar radiation leaving the galaxy unattenuated). Massive stars being embedded in molecular cocoons, they have a lower escape fraction than Low- and Intermediate-
<figure><img src="image_336.png" /></figure>

*Mass Stars* (LIMS) which occupy lower density regions. In the *DustPedia* sample, we showed that the escape fraction was on average ≈ 81%, with mild variations across galaxy types (Bianchi et al., 2018). It is slightly lower in LTGs (≈ 75%). We emphasize that this nearby galaxy sample lacks the deeply enshrouded star formation of LIRGs and ULIRGs, where the global escape fraction can drop down to ≈ 1% (e.g. Clements et al., 1996). The question of the dust heating contribution can now be tackled with more accuracy using 3D MCRT models.

**Large-scale radiative transfer models of galaxies.** Applying a 3D MCRT model to reproduce the spatial flux distribution of galaxies, in all wavebands, is not straightforward. Indeed, the observations provide only 2D projected constraints. This is why most studies favor edge-on galaxies, as the images of such objects provide constraints on both the radial and azimuthal distributions, assuming axisymmetry (Fig. III.25). Several studies have modeled the effect of extinction on the optical data of disk galaxies using such codes (e.g. Xilouris et al., 1999; Alton et al., 2004; Bianchi, 2007). They were able to answer the recurring question about the optical thickness of disk galaxies (Disney et al., 1989). In particular, Xilouris et al. (1999) found that the face-on optical depth of typical spiral galaxies is less than one, in all optical bands. Concerning dust heating, recent progress has been made, especially by the *DustPedia* collaboration, using the MCRT code SKIRT (Baes & Camps, 2015).

* In M 31, Viaene et al. (2017) showed that 90% of the dust could be heated by the evolved stellar populations.

* Nersesian et al. (2020a) showed that dust heating by young stars accounts on average for ≈ 60% in four face-on barred spirals.

* Viaene et al. (2020) modeled the strong AGN NGC 1068. We showed that ≈ 80% of the heating is coming from young stars, and only a few percents from the central engine. However, dust heating by AGN represent about ≈ 90% within the central ≈ 100 pc.

* Nersesian et al. (2020b) studied M 51, showing that globally ≈ 71% of the heating was coming from young stars. Surprisingly, we also found that NGC 5195, the companion of M 51 was responsible for ≈ 6% of this heating.

A model such as SKIRT can also be used to model the radiative transfer in simulations of galaxies (e.g. Trčka et al., 2020). Finally, these models account for the energy balance between the escaping UV-visible light and the re-emitted IR-submm radiation. Interestingly enough, several studies report a deficit of modeled FIR emission by a factor ≈ 3 – 4, compared to the observations (Alton et al., 2000, 2004; Dasyra et al., 2005; De Looze et al., 2012a,b). This discrepancy is thought to result from a lack of detail in modeling the geometry. In particular, the presence of young stars, deeply embedded in molecular clouds, at sub-grid resolutions, could compensate for this deficit without significantly altering the extinction (cf. e.g. Baes et al., 2010).

### III.1.3.3 Constraining the Grain Opacity

Dust masses derived from SED fits directly depend on the assumed grain opacity. Using the MBB parametrization (Eq. I.73), both the scaling, $\kappa_0$, and the emissivity index, $\beta$, are important. There are particular situations, where we can reverse the process and constrain these two parameters:

1. if we are observing a region that we can assume uniformly illuminated, then we can infer $\beta$;

2. if we have an independent constraint on the dust mass, then we can infer $\kappa_0$.

**Studies of the emissivity index.** There are numerous publications presenting MBB fits of nearby galaxies. However, as discussed in Sect. III.1.2.1, the derived emissivity index, $\beta$, is degenerate with temperature mixing. The best constraints on the intrinsic $\beta$ are obtained in the submm regime, where only massive amounts of very cold dust (T $\lesssim$ 10 K) could bias the value. Table III.3 lists *effective emissivity indices*, $\beta_{eff}$, for several objects, obtained with *Planck*, with constraints up to 850 µm. It

<figure><img src="image_337.png" /></figure>
<figure><img src="image_338.png" /></figure>

<figure><img src="image_339.png" /><figcaption>FIGURE III.25 – Radiative transfer modeling of NGC 4565. The observations (left column) are compared to the modeled flux distribution (right column). Credit: De Looze et al. (2012a).</figcaption></figure>

appears that all the values are lower than 2, and that low-metallicity systems have a lower $\beta_{eff}$ than higher metallicity galaxies. Boselli et al. (2012), studying a volume-limited sample with Herschel (up to 500 µm), also found an average $\beta_{eff} \approx 1.5$, and hinted that low-metallicity objects tend to have $\beta_{eff} < 1.5$. In M 33, $\beta_{eff}$ derived from Herschel observations is around 2 in the center and decreases down to 1.3 in the outer parts (Tabatabaei et al., 2014). On the other hand, the outer regions of M 31 exhibit a steeper slope ($\beta_{eff} \approx 2.3$) than in its center (Draine et al., 2014). This contradictory behaviour does not appear to originate in fit biases, as both increasing and decreasing trends of $\beta_{eff}$ with radius are found in the sample of Hunt et al. (2015).

<table><caption>TABLE III.3 – Free emissivity index MBB fits of nearby galaxies by the Planck collaboration.</caption><thead><tr><th></th><th>Milky Way</th><th>M 31</th><th>LMC</th><th>SMC</th></tr></thead><tbody><tr><th>Temperature</th><td>19.7 ± 1.4 K</td><td>18.2 ± 1.0 K</td><td>21.0 ± 1.9 K</td><td>22.3 ± 2.3 K</td></tr><tr><th>β<sub>eff</sub></th><td>1.62 ± 0.10</td><td>1.62 ± 0.11</td><td>1.48 ± 0.25</td><td>1.21 ± 0.27</td></tr><tr><th>Reference</th><td>Planck (2014a)</td><td>Planck (2015a)</td><td>Planck (2011a)</td><td>Planck (2011a)</td></tr></tbody></table>

**Grain opacity in the LMC.** An important result, that has often been misunderstood, concerns the grain opacity in the LMC. Galliano et al. (2011), modeling a strip covering one fourth of the LMC (cf. Fig. III.21), with the composite approach (Sect. III.1.2.2), found that the dustiness distribution in this galaxy was most of the time larger than the maximum value it could in principle reach. This maximum value is set by the elemental abundances. The fraction of elements locked-up in grains can indeed not be larger than the amount available in the ISM. The metallicity of the LMC is $Z_{LMC} \approx 1/2 Z_{\odot}$ (Pagel, 2003). The maximum dustiness of the LMC is thus $Z_{dust}^{max} \approx Z_{LMC} \approx 0.007$. The dust mixture that was used is an update of the Zubko et al. (2004, BARE-GR-S). It is essentially based on the Draine & Li (2007) optical properties, a pre-Herschel model. This discrepancy is shown in Fig. III.26 (the red histogram). The only explanation is that this grain mixture is not emissive enough to account for the observed FIR-submm emission. We thus proposed an alternate dust model, simply replacing graphite by amorphous carbons (the ACAR sample of Zubko et al., 1996), without altering the total carbon fraction. This simple modification boosts the emissivity by a factor $\approx 2-3$. With this new model, most of the dustiness distribution is centered around its expected value, and is clear from

<footnote n="6">6. This factor is not precise, as the slope of the opacity is also changed. The difference in emissivity is thus not a sole scaling.</footnote>
<figure><img src="image_340.png" /></figure>

the forbidden range (yellow). It was called the *AC model* (blue histogram in Fig. III.26). The tail of the distribution in the forbidden zone originates in cold regions, where the uncertainty is large. The conclusion of this modeling was that LMC grains must be a factor ≈ 2–3 more emissive than the Draine & Li (2007) model. We actually presented a preliminary version of this result during Herschel's science demonstration phase (Meixner et al., 2010).

<figure><img src="image_341.png" /></figure>

FIGURE III.26 – Dust mass discrepancy in the LMC. The histograms are the pixel distribution of the dustiness in the LMC strip of Galliano et al. (2011). We show two models: (i) the “standard model” (red), based on the Draine & Li (2007) optical properties; (ii) the “AC model” (blue), replacing graphite by amorphous carbon to boost the emissivity. Most pixels of the standard model are in the hatched area, thus violating the constraints put by heavy element abundances. We also show the uncertainties on the limit dustiness (yellow error bar) and on the expected one (green error bar). Licensed under CC BY-SA 4.0.

**Confirmation in other systems.** The conclusion of the Galliano et al. (2011) study was that grains in the LMC had to be more emissive than in the MW, because the Draine & Li (2007) model was at the time consistent with the MW. This last statement however happened to be inexact. Planck Collaboration et al. (2016c) modeled the all sky dust emission, using also the Draine & Li (2007) model. The A(V) estimated along the sightlines of ≳ 200 000 Quasi-Stellar Objects (QSO) was systematically lower than their dust-emission-derived A(V). Their comparison of emission and extinction thus indicates that the Galactic opacity is in fact also a factor of ≈ 2 higher than previously assumed. In addition, in M 31, Dalcanton et al. (2015) derived a high spatial resolution map of A(V). As in the Galaxy, the emission-derived A(V) map (Draine et al., 2014) was found to be a factor of ≈ 2.5 higher. We emphasize that, although each of these studies found evidence of local variations of the emissivity as a function of the density (cf. Sect. IV.2.1.1), the overall opacity seems to be scaled up compared to Draine & Li (2007). In other words, in all the environments where enough data is available to constrain κ, it is found a factor of ≈ 2–3 higher than the original Draine & Li (2007) properties. Dust models therefore need to use an opacity consistent with these constraints. This is the case of the THEMIS model. Its FIR-submm opacity is very close to our AC model (cf. Fig. 4 of Galliano et al., 2018).

<figure><img src="image_342.png" /></figure>
☞ It is reasonable to adopt the THEMIS grain opacity (cf. Sect. II.3.3), when modeling galaxies.

**The opacity in nearby galaxies.** The DustPedia collaboration conducted several studies aimed at constraining the grain opacity in nearby galaxies. First, Bianchi et al. (2019) studied the actual emissivity of 204 late-type galaxies, that is the ratio of IR emission to H column density. We found an emissivity $\epsilon_v$(250 µm) ≈ 0.82 ± 0.07 MJy/sr/(1020 H/cm²), consistent with the MW, except for the hottest sources. These estimates were derived using global fluxes, integrated over the whole galaxy. In parallel, Clark et al. (2019) modeled in details the two face-on galaxies, M 74 and M 83. We could map the grain opacity. This was done by converting metallicity maps into oxygen depletion maps, and comparing those to the dust mass. The derived opacities were quoted at λ = 500 µm: κ(500 µm) ≈ 0.11–0.25 m²/kg in M 74, and κ(500 µm) ≈ 0.15–0.80 m²/kg in M 83. These values are consistent with the Herschel-Planck-revised opacities of the THEMIS model (κ(500 µm) ≈ 0.19 m²/kg; cf. Fig. II.26).

### III.1.3.4 Constraining the Size Distribution

We have seen in Sect. III.1.2.2 that there is a degeneracy between the grain size and starlight intensity distributions. This degeneracy arises from the fact that it is observationally difficult to distinguish the MIR emission of a hot region from the MIR emission of small grains. In the early 2000s, we did not know it was impossible, so we did it (Galliano et al., 2003, 2005). We modeled the SED of the following four BCDs: NGC 1569 (Z ≈ 1/4 Z☉), II Zw 40 (Z ≈ 1/6 Z☉), He 2-10 (Z ≈ 1 Z☉) and NGC 1140 (Z ≈ 1/3 Z☉), to infer their size distribution. We interpreted these results in light of shock processing.

**Grain processing by shock waves.** Shock waves from SNe process dust grains, while sweeping the ISM.

* At high velocity, $v_{\text{shock}} \gtrsim 1000$ km/s, they vaporize most of the dust (e.g. Dwek, 1998; Slavin et al., 2015).

* At intermediate velocity, $50 \text{ km/s} \lesssim v_{\text{shock}} \lesssim 200 \text{ km/s}$, shattering and fragmentation become dominant, altering the size distribution (e.g. Jones et al., 1996; Bocchio et al., 2014).

Fig. III.27 shows the model of Jones et al. (1996) for graphite and silicates. In both panels, the grey curve is the initial, MRN size distribution. The color curves show the size distribution obtained after the mixture has been swept by a shock of velocity, $v_{\text{shock}}$. The main effect of the blast wave is to fragment and shatter grains, turning large grains into smaller grains. The qualitative effect is similar for both compositions. This is best seen at low velocity (blue curves). Large grains are depleted and there is an excess of small grains. At higher velocity, the distribution tends toward a log-normal centered around $a \approx 10$ nm (magenta curve). Vaporization also leads to a net loss of dust mass. This model has since then been refined by Bocchio et al. (2014), who applied it to THEMIS grains. We will discuss more extensively dust processing by SN blast waves in Chap. IV.

**The modeling strategy.** Galliano et al. (2003, 2005) modeled the UV-to-mm global SED of the four BCDs, using the DBP90 dust model, the stellar evolutionary synthesis code PÉGASE (Fioc & Rocca-Volmerange, 1997, Fig. III.19), and the photoionization code, CLOUDY (Ferland et al., 1998). The modeling scheme was the following.

1. We assume that the ISM is concentrated in a spherical shell. At the center of this shell are the stellar populations.

2. We model the escaping UV-visible SED with two PÉGASE stellar populations, a young and an old one.
<figure><img src="image_343.png" /><figcaption>FIGURE III.27 – *Effect of a blast wave on the grain size distribution.* In both panels, we demonstrate the processing of a MRN size distribution (grey) by shock waves of different velocities, $v_{\text{shock}}$. Panel (a) shows graphite, and panel (b), silicates, both from the Jones et al. (1996) model. Licensed under CC BY-SA 4.0.</figcaption></figure>

3. We further constrain the UV-visible fit by selecting intrinsic stellar spectra that have the appropriate hardness. The hardness of the ISRF is constrained by matching the observed ISO [NeIII]15.56µm/[NeII]12.81µm and [SIV]10.51µm/[NeIII]15.56µm ratios, using CLOUDY.

4. The intrinsic ISRF is then used to heat the DBP90 dust mixture. We fit the dust emission to the IR-to-mm observations, by varying the size distribution.

5. We iterate this process a few times, as the dust size distribution impacts the extinction curve, which is circularly used to constrain the stellar populations.

This model is self-consistent *per se*. It however avoids the degeneracy between size and ISRF distributions by assuming a simple geometry (the shell). The dust is thus uniformly illuminated in this model, which is an unrealistic assumption for a star-forming galaxy. It is therefore likely that some of the emission we have attributed to small grains originates in hot regions. The results Galliano et al. (2003, 2005) obtained, that we will discuss in the following paragraphs, are nevertheless qualitatively consistent with the properties we expect in these environments. It is possible that only a fraction of the MIR emission originates in compact HII regions, not excluding an overabundance of small grains.

**The grain size distribution in four dwarf galaxies.** The inferred size distribution, in the four BCDs, are shown in Fig. III.28. We display the three components of the DBP90 model: PAHs, VSGs and BGs (cf. Sect. II.1.2.5). The most striking features, common to the four objects, are the following.

* PAHs are under-abundant, compared to the MW. This was one of the first attempts at constraining the PAH abundance in low-metallicity environments. Nowadays, there are overwhelming evidence of a general trend between the aromatic feature strength and metallicity. We will come back to this point in Sect. IV.2.2.1.

* The global grain size distribution is dominated by small grains with radii of a few nanometers. These size distributions are thus qualitatively consistent with shock-processed grains. This is encouraging, as we know that these environments are permeated by numerous SN blast waves, coming from their young stellar populations (e.g. Oey, 1996; Izotov et al., 2007).
<figure><img src="image_344.png" /></figure>

Two studies used a similar approach, assuming uniform illumination, and fitting the SED varying the size distribution of the DBP 90 model, in NGC 1569 (Lisenfeld et al., 2002) and in the LMC (Paradis et al., 2009). They also concluded to an overabundance of small grains.

<figure><img src="image_345.png" /><figcaption>FIGURE III.28 – Grain size distribution in four dwarf galaxies. In each panel, we show the MW grain size distribution of the DBP 90 model, as dashed lines. Those are the three components presented in Sect. II.1.2.5: PAH, VSG and BG. The filled curves show the likely range of size distribution inferred by the SED fits of Galliano et al. (2003, 2005). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Consequence on the extinction curves.** We have briefly mentioned in Sect. II.1.2.2 that the extinction curves in the Magellanic clouds were systematically different from the MW. Fig. III.29.a compares the extinction towards different sightlines in the LMC and SMC, to the range of extinction curves in the MW. We see that the LMC (Z ≈ 1/2 Z⊙) is on average (red curve) similar to the MW. However, toward the massive star-forming region 30 Doradus (LMC2 supershell; green curve), it is steeper, with a weaker 2175 Å bump. When we go to the SMC (Z ≈ 1/5 Z⊙; blue curve), the difference is more pronounced: the extinction curve has a very steep UV-rise and lacks the 2175 Å bump. The origin of these

<figure><img src="image_346.png" /></figure>
<figure><img src="image_347.png" /></figure>

variations are still debated. Our four BCDs brought an interesting perspective on this open question. The extinction curves derived from the size distributions of Fig. III.28 are shown in Fig. III.29.b. We can see that they are systematically steeper than the average MW (R(V) = 3.1). They also have a 2175 Å that is either weaker (NGC 1569, He 2-10 and NGC 1140) or similar (II Zw 40) to the MW<fnref n="7" />. In other words, these extinction curves lie between the LMC and SMC, consistent with the metallicity range of these BCDs. The modeling of Galliano et al. (2003, 2005) therefore provides a coherent view of the grain properties in these environments, where shock waves have an instrumental role in shaping the grain sizes.

<figure><img src="image_348.png" /></figure>

FIGURE III.29 – Extinction curves of low-metallicity environments. In both panels, we show as a reference the range of Galactic extinction curves, with R(V) = 2–5, from the Fitzpatrick et al. (2019) sample (cf. Fig. II.11), in grey. The white curve is the average Galactic extinction, with R(V) = 3.1. In panel (a), we overlay several extinction curves within the Magellanic clouds, from the Gordon et al. (2003) sample: (i) the average of the LMC (red); (ii) the LMC2 supershell (green), near the massive star-forming region 30 Doradus; and (iii) the SMC bar (blue). In panel (b), we show the extinction of the four BCDs modeled by Galliano et al. (2003, 2005). Licensed under CC BY-SA 4.0.

## III.2 Studies Focussing on Specific Spectral Domains

Sect. III.1 was devoted to modeling the whole IR SED, which is necessary to estimate the total dust content of galaxies. There are however several other important properties that can be self-consistently studied by focussing on a particular wavelength range.

### III.2.1 Scrutinizing Mid-IR Spectra

Mid-IR spectra have been extensively observed since the first light of ISO. Spitzer and AKARI have extended our knowledge of this spectral range and the JWST will likely revolutionize it.

<footnote n="7">7. We note that the strength of the bump is controlled by the carbon-to-silicate grain ratio, a parameter that is poorly constrained by the SED fit.</footnote>
### III.2.1.1 The Aromatic Feature Spectrum

Until now, we have discussed PAHs from a general point of view, and how to estimate their mass fraction. We now focus on the information that the analysis of their detailed MIR emission spectrum can bring. In what follows, we interchangeably use the terms UIBs, aromatic features and PAH bands. The only case where these terms are not equivalent is when discussing the UIBs around 3 µm, coming from a mixture of aromatic and aliphatic features, that can not be solely attributed to PAHs.

**MIR spectra of galaxies.** Fig. III.30 illustrates the diversity of MIR spectra encountered in different environments. It shows three extreme galaxies from the Hu et al., in prep. sample.

**Gas-rich, Solar-metallicity galaxies,** such as NGC 1097 (green spectrum in Fig. III.30), have bright aromatic features. These features are emitted by both their diffuse ISM and their PDRs. The level and steepness of their MIR continuum, longward 10 µm, depend on their SFR. Galaxies with a low SF-activity tend to have a flatter continuum, as the emission from hot equilibrium grains in HII regions is lower.

**Low-metallicity galaxies,** such as NGC 1569 (blue spectrum in Fig. III.30), have an integrated spectrum very similar to an HII region (e.g. Peeters et al., 2002b; Martín-Hernández et al., 2002). They have weak or undetected aromatic features, which we will extensively discuss in Sect. IV.2.2.1. Their strong ionic lines result from the combination of their young stellar population and lower dust screening. For the same reason, their MIR continuum is significantly steeper than Solar-metallicity objects having the same specific Star Formation Rate (sSFR ≡ SFR/M<sub>⋆</sub>). In addition to small, stochastically-heated grains, the MIR continuum of BCDs originates partly in large grains at thermal equilibrium with the radiation field in HII regions. The latter grains are indeed hot enough to significantly emit at shorter wavelengths than Solar-metallicity objects.

**AGNs,** such as CentaurusA (red spectrum in Fig. III.30), have a rather flat NIR-to-MIR continuum (e.g. Laurent et al., 2000; Spoon et al., 2007). This continuum originates in the strong gradient of temperature due to the central illumination by the AGN (this is the same qualitative effect as in our MCRT simulation; Sect. III.1.1.4). In particular, temperatures of the underlying continuum can reach higher values in the central region, than normal galaxies. It explains why the NIR continuum does not drop to zero. These central hot regions produce a bright MIR continuum, attenuated by the entire disk, resulting in deep silicate absorption bands.

**Laboratory and theoretical PAH physics.** Although we do not know the exact composition of the interstellar carbon grain mixture responsible for the aromatic feature emission, the brightest bands have been attributed to the main vibrational modes of PAHs. There are still some debates about the origin of the weakest features (cf. e.g. Allamandola et al., 1999; Verstraete et al., 2001; Tielens, 2008; Boersma et al., 2010; Jones et al., 2013). In Fig. III.30, we have labeled the different bands with a given mode. These modes are schematically represented in Fig. III.31.a.

**The charge** of the molecules is one of the most important parameters controlling the ratio between the C–H and C–C bands. Fig. III.32.a shows the laboratory data of Allamandola et al. (1999). The two spectra are the sum of several neutral and cationic molecules. It is clear that C–C and C–C–C bands (6.2, 7.7 and 17 µm complexes) are predominantly carried by ionized PAHs, whereas C–H bands (3.3, 11.3 and 12.7 µm complexes) are carried essentially by neutral PAHs.

**Dehydrogenation** has a similar effect to ionization. However, for PAHs with more than ≈ 25 C atoms (i.e. the bulk of interstellar PAHs), hydrogenation through reactions with abundant atomic H is more important than H loss through unimolecular dissociation (cf. e.g. Hony et al., 2001). Thus, dehydrogenation does not have a detectable effect on the UIB spectrum.

**The molecular structure** is another factor. C–H *Out-Of-Plane* (OOP) bending modes have different frequencies, depending on the number of H atom per aromatic cycle (cf. Fig. III.31.b). The
<figure><img src="image_349.png" /><figcaption>FIGURE III.30 – *MIR spectra of galaxies.* We show the AKARI and *Spitzer* spectra of three galaxies: (i) CentaurusA (red) is an ETG with a powerful AGN; (ii) NGC 1097 (green) is a LTG with a low-luminosity AGN; and (iii) NGC 1569 (blue) is BCD that we have scaled down by a factor of 3. These spectra were prepared for Galliano et al. (2018). We have indicated most of the relevant features. Silicate absorption bands are shown in grey, and ice absorption bands in cyan (cf. Sect. II.2.1.2). Gas-phase emission lines are indicated in orange, and carbon-dust emission features, in magenta. Licensed under CC BY-SA 4.0.</figcaption></figure>
11.3 µm band corresponds to a solo H, found on straight molecular edges, whereas the 12.7 µm one corresponds to a trio, found on corners of the molecules. The solo-to-trio intensity ratio, I<sub>11.3</sub>/I<sub>12.7</sub>, is thus an indicator of PAH compactness (I<sub>λ</sub> being the integrated intensity of the feature centered at λ µm).

**The size** of the PAHs affects the relative intensity of the different bands. This is demonstrated in Fig. III.32.b. Small PAHs (magenta and red spectra) fluctuate up to temperatures higher than large ones (blue and cyan spectra). Short-wavelength bands are therefore more pumped in small PAHs, whereas large PAHs emit predominantly long-wavelength features.

**ISRF hardness** has an effect similar to the size, as a higher mean photon energy causes the grain to fluctuate up to higher temperatures (Sect. III.2.1.3 and e.g. Galliano et al., 2008b). We have however seen, in Fig. III.14, that this effect was probably less than a factor ≈ 2, in astrophysically relevant cases.

The charge and size of the PAHs are the main parameters controlling their emission spectrum.

<figure><img src="image_350.png" /><figcaption>FIGURE III.31 – Vibrational modes of PAHs. Panel (a) represents the main in-plane and out-of-plane vibrational modes. Panel (b) gives examples of solo, duo, trio and quartet H sites. In both panels, red spheres represent C atoms, and cyan spheres, H atoms. Licensed under CC BY-SA 4.0.</figcaption></figure>

### III.2.1.2 Spectral Decomposition Methods

To study variations of the aromatic feature spectrum with environmental conditions, one needs to measure the intensity of each band. This task is not as simple as it appears. There are indeed several challenges.

* Aromatic bands are broad features with large wings. They are usually modeled with Lorentz or Drude profiles (e.g. Smith et al., 2007; Galliano et al., 2008b; Lai et al., 2020). Their intensity drops quite slowly, away from their peak. Consequently, deblending adjacent features and distinguishing features from the continuum is difficult.

* The exact number and spectral shape of the different aromatic features is not precisely known (e.g. Verstraete et al., 1996; Boulanger et al., 1998). High-spectral-resolution, high-signal-to-noise-ratio spectra allow us to identify up to ≈ 40 individual bands, some being very weak. The
<figure><img src="image_351.png" /></figure>

<figure><img src="image_352.png" /><figcaption>FIGURE III.32 – Laboratory and theoretical PAH spectra. Panel (a) shows the absorption coefficient of neutral and cationic PAHs measured in the laboratory by Allamandola et al. (1999). Panel (b) shows theoretical emission spectra of neutral and cationic PAHs, with radii a = 0.35 nm and a = 5 nm. We have used the Draine & Li (2007) optical properties and computed the stochastic heating for the Solar neighborhood ISRF (U = 1). Licensed under CC BY-SA 4.0.</figcaption></figure>

assumption about the number and spectral properties of these bands therefore adds another layer of uncertainties.

* Absorption by silicates and ices introduce a degeneracy between A(V) and the band strength. In particular, unless the 9.8 µm band is deep enough to be unambiguously constrained (such as in Cen A; Fig. III.30), the 8 – 12 µm region can equivalently be modeled with: (i) 8.6 µm and 11.3 µm bands and no extinction; or (ii) weaker 8.6 µm and 11.3 µm bands, a higher underlying continuum and some extinction.

MIR spectral decomposition methods are therefore an essential tool to properly study PAH features.

**Calibrating feature properties.** Since the properties of the different *UIBs* are not *a priori* known, we need to empirically infer them. In our work, we parametrize the band spectral profile with a *split-Lorentz function* (Hu et al., 2021a):

$$F_v = I \times \begin{cases} \frac{2 \Delta v_s^2}{\pi \Delta v_s + \Delta v_l (v - v_0)^2 + \Delta v_s^2} & \text{for } v \ge v_0 \\ \frac{2 \Delta v_l^2}{\pi \Delta v_s + \Delta v_l (v - v_0)^2 + \Delta v_l^2} & \text{for } v < v_0, \end{cases} \quad (III.42)$$

where $v_0$ is the central frequency of the feature, and $\Delta v_s$ and $\Delta v_l$ are its widths on the short- and long-wavelength sides. Having an asymmetric feature is indeed necessary to accurately fit good quality spectra. This asymmetry may originate in the anharmonicity of the transition responsible for the band, or may be due to unresolved blended features. The parameters characterizing each individual features, $v_0$, $\Delta v_s$ and $\Delta v_l$, could be derived from each fit. However, most of them would be quite uncertain, using an average *Spitzer* spectrum. For that reason, we have calibrated these parameters (i.e. inferred their reference value), using high-resolution, high-signal-to-noise-ratio spectra of Galactic regions. This calibration is demonstrated in Fig. III.33.

<figure><img src="image_353.png" /></figure>
<figure><img src="image_354.png" /></figure>

1. We have first fitted the ISO/SWS spectrum of the reflection nebula, the *Red Rectangle*, in order to calibrate the narrow bands (cf. Fig. III.33.a). The very wide bands, also called *plateaus*, are not very prominent in this region.

2. We then fix the narrow band parameters and infer the $\lambda \approx 8$ µm and $\lambda \approx 12$ µm plateau parameters by fitting the ISO/SWS spectrum of the planetary nebula, NGC 7027 (cf. Fig. III.33.b). The $\lambda \approx 17$ µm complex is too weak to be calibrated using this spectrum.

3. The $\lambda \approx 17$ µm complex is calibrated by fitting the IRS spectrum of the PDR, M 17 (cf. Fig. III.33.c).

4. The aromatic and aliphatic bands in the $\lambda \approx 3$ µm region are calibrated by fitting the $\lambda = 3 - 5$ µm spectrum of NGC 7027 (cf. Fig. III.33.d). The continuum in this range is indeed clean and there are no other blended bands.

Table III.4 gives the resulting band parameters. With these parameters fixed, we can fit even low-signal-to-noise-ratio spectra varying only the intensity of each band (parameter I in Eq. III.42).

**Fitting every feature at once.** The total MIR spectrum contains the emission from different physical processes that have to be separated in order to accurately measure UIB intensities (cf. Fig. III.30). Several models have tackled this problem since the ISO days (e.g. Verstraete et al., 1996; Boulanger et al., 1998; Laurent et al., 2000; Madden et al., 2006; Smith et al., 2007; Galliano et al., 2008b; Mori et al., 2012; Lai et al., 2020).

**Individual UIBs** can be fitted with Lorentz profiles (e.g. Galliano et al., 2008b), Drude profiles (e.g. Smith et al., 2007) or split-Lorentz profiles (cf. Table III.4 and Fig. III.33; e.g. Hu et al., in prep.).

**The dust continuum** can be fitted with a linear combination of grey bodies. The opacity of these grey bodies can be a simple MBB power-law (e.g. Smith et al., 2007) or the opacity of realistic materials (e.g. Hu et al., in prep.). This dust continuum likely originates in a combination of small, stochastically-heated grains, and large, hot equilibrium grains. The fitted parameters (mass and temperature) of each grey body therefore have to be considered as nuisance variables. They can not be interpreted physically, because of the uncertainty of their heating mechanism. They are employed only to decompose the continuum and UIB emissions.

**Gas lines** can be fitted with Gaussian profiles. Table III.5 lists the most prominent MIR gas lines and their central wavelengths. The width of the profile is determined by the spectral resolution of the instrument, except at very high spectral resolution ($R \equiv \lambda/\Delta\lambda \gtrsim 30000$), where the true width of the line can be resolved.

**The stellar continuum** only contributes at short wavelength, and can be modeled with a Rayleigh-Jeans law.

**The extinction**, which is essentially absorption at these wavelengths, can be modeled with a general extinction curve (cf. e.g. Fig. II.24), assuming a simple geometry (cf. e.g. Sect. III.1.1.2). The most prominent dust features are the two silicate bands at 9.8 and 18 µm (cf. Fig. III.30). Ice absorption can also be non negligible. They can be taken into account the same way, as icy band profiles are well-constrained (cf. Fig. II.12.b). The optical depth ratio between different ice species and the silicates however varies from one source to the other (e.g. Yamagishi et al., 2015). The optical depth therefore has to be independently estimated for each species (i.e. one needs to derive $\tau_{dust}$, $\tau_{H_2O}$, $\tau_{CO}$, etc.).

Fig. III.34.a demonstrates such a fitting method on the total spectrum of M 82 (Galliano et al., 2008b). It is labeled as the *Lorentzian method*, as the UIBs are modeled with Lorentz profiles. This earlier model did not use all the bands given in Table III.4, that we are now using.

**Alternative methods.** Several other MIR spectral fitting methods have been discussed in the literature. The two following ones are worth mentioning.

<figure><img src="image_355.png" /></figure>
<figure><img src="image_356.png" /></figure>

<table><caption>TABLE III.4 – *UIB profile parameters*. These are the parameters of Eq. (III.42). We have converted the frequencies in wavelengths: $v_0 \equiv c/\lambda_0$, $\Delta v_s \equiv c/(\lambda_0 - \Delta\lambda_s) - c/\lambda_0$, $\Delta v_l \equiv c/\lambda_0 - c/(\lambda_0 + \Delta\lambda_l)$. These are the parameters used for the work of Hu et al. (in prep.).</caption><thead><tr><td>λ<sub>0</sub></td><td>Δλ<sub>s</sub></td><td>Δλ<sub>l</sub></td><td>Type</td></tr></thead><tbody><tr><td>3.291 µm</td><td>0.020 µm</td><td>0.019 µm</td><td>Main</td></tr><tr><td>3.399 µm</td><td>0.011 µm</td><td>0.024 µm</td><td>Main</td></tr><tr><td>3.499 µm</td><td>0.077 µm</td><td>0.071 µm</td><td>Small</td></tr><tr><td>5.239 µm</td><td>0.025 µm</td><td>0.058 µm</td><td>Small</td></tr><tr><td>5.644 µm</td><td>0.040 µm</td><td>0.080 µm</td><td>Small</td></tr><tr><td>5.749 µm</td><td>0.040 µm</td><td>0.080 µm</td><td>Small</td></tr><tr><td>6.011 µm</td><td>0.040 µm</td><td>0.067 µm</td><td>Small</td></tr><tr><td>6.203 µm</td><td>0.031 µm</td><td>0.060 µm</td><td>Main</td></tr><tr><td>6.267 µm</td><td>0.037 µm</td><td>0.116 µm</td><td>Main</td></tr><tr><td>6.627 µm</td><td>0.120 µm</td><td>0.120 µm</td><td>Small</td></tr><tr><td>6.855 µm</td><td>0.080 µm</td><td>0.080 µm</td><td>Small</td></tr><tr><td>7.079 µm</td><td>0.080 µm</td><td>0.080 µm</td><td>Small</td></tr><tr><td>7.600 µm</td><td>0.480 µm</td><td>0.502 µm</td><td>Plateau</td></tr><tr><td>7.617 µm</td><td>0.119 µm</td><td>0.145 µm</td><td>Main</td></tr><tr><td>7.870 µm</td><td>0.170 µm</td><td>0.245 µm</td><td>Main</td></tr><tr><td>8.362 µm</td><td>0.016 µm</td><td>0.016 µm</td><td>Small</td></tr><tr><td>8.620 µm</td><td>0.183 µm</td><td>0.133 µm</td><td>Main</td></tr><tr><td>9.525 µm</td><td>0.107 µm</td><td>0.600 µm</td><td>Small</td></tr><tr><td>10.707 µm</td><td>0.100 µm</td><td>0.100 µm</td><td>Small</td></tr><tr><td>11.038 µm</td><td>0.027 µm</td><td>0.073 µm</td><td>Small</td></tr><tr><td>11.238 µm</td><td>0.053 µm</td><td>0.153 µm</td><td>Main</td></tr><tr><td>11.400 µm</td><td>0.720 µm</td><td>0.637 µm</td><td>Plateau</td></tr><tr><td>11.796 µm</td><td>0.021 µm</td><td>0.021 µm</td><td>Small</td></tr><tr><td>11.950 µm</td><td>0.080 µm</td><td>0.222 µm</td><td>Small</td></tr><tr><td>12.627 µm</td><td>0.200 µm</td><td>0.095 µm</td><td>Main</td></tr><tr><td>12.761 µm</td><td>0.081 µm</td><td>0.140 µm</td><td>Main</td></tr><tr><td>13.559 µm</td><td>0.160 µm</td><td>0.161 µm</td><td>Small</td></tr><tr><td>14.257 µm</td><td>0.152 µm</td><td>0.059 µm</td><td>Small</td></tr><tr><td>15.893 µm</td><td>0.178 µm</td><td>0.200 µm</td><td>Small</td></tr><tr><td>16.483 µm</td><td>0.100 µm</td><td>0.059 µm</td><td>Small</td></tr><tr><td>17.083 µm</td><td>0.496 µm</td><td>0.562 µm</td><td>Plateau</td></tr><tr><td>17.428 µm</td><td>0.100 µm</td><td>0.100 µm</td><td>Small</td></tr><tr><td>17.771 µm</td><td>0.031 µm</td><td>0.075 µm</td><td>Small</td></tr><tr><td>18.925 µm</td><td>0.037 µm</td><td>0.116 µm</td><td>Small</td></tr></tbody></table>

<figure><img src="image_357.png" /></figure>
<figure><img src="image_358.png" /></figure>

<figure><img src="image_359.png" /><figcaption>FIGURE III.33 – *Empirical calibration of UIB profiles.* In each panel, the black dots with error bars (barely visible) represent the observations. These are ISO/SWS spectra except for M 17, which is a *Spitzer/IRS* spectrum. The fitted model is the sum of the different individual components: (i) grey bodies for the dust continuum; (ii) Gaussian profiles for gas lines; and (iii) split-Lorentz profiles (Eq. III.42) for UIBs. Licensed under CC BY-SA 4.0.</figcaption></figure>

**The spline method** consists in interpolating the spectrum under the main band complexes, using spline functions (e.g. Vermeij et al., 2002; Galliano et al., 2008b). It is demonstrated in Fig. III.34.b. A first spline interpolation defines the continuum and a second one defines the band plateaus. The areas between the two continua (magenta filled) represent the plateaus in Fig. III.34.b. The bands are then simply the intensity above the plateaus (cyan filled). The advantages of this method are: (i) it is simple to implement; (ii) it is very fast to run; and (iii) it does not suffer from the degeneracies between blended features, or between band wings and the continuum. It has however several limitations: (i) the choice of the spline anchor points is arbitrary, which results in systematic differences with other methods; (ii) at medium-spectral resolution (typical of ISO/CAM or *Spitzer/IRS* SL-LL modes) it is impossible to deblend features and lines, such as the 12.7 µm UIB and the [NeII]$_{12.81µm}$ line; and (iii) this method does

<figure><img src="image_360.png" /></figure>
<figure><img src="image_361.png" /></figure>

<figure><img src="image_362.png" /></figure>

FIGURE III.34 – *MIR spectral fitting methods.* Demonstration of the three MIR fitting methods, described in this manuscript, on the total ISO/CAM spectrum of the starburst galaxy, M 82 (Galliano et al., 2008b). The observations (the black error bars) are identical in each panel. The models of panels (a) and (c) are least-squares fits, whereas panel (b) is a spline interpolation. Panels (a) and (b) are from Galliano et al. (2008b). The fit of panel (c) has been performed only up to $\lambda = 14$ µm, using PAHtat (Pilleri et al., 2012). In addition to the four components of Fig. III.35, a continuum and the main gas lines are simultaneously fitted. Licensed under CC BY-SA 4.0.

<figure><img src="image_363.png" /></figure>
<table><caption>TABLE III.5 – Most prominent MIR gas lines.</caption><thead><tr><th>Central wavelength</th><th>Species</th><th>Transition</th><th>Type</th></tr></thead><tbody><tr><td>4.052 µm</td><td>H<sub>1</sub></td><td>Brackett α</td><td>recombination</td></tr><tr><td>5.511 µm</td><td>H<sub>2</sub></td><td>0–0 S(7)</td><td>ro-vibrational</td></tr><tr><td>5.908 µm</td><td>H<sub>1</sub></td><td>Humphreys γ</td><td>recombination</td></tr><tr><td>6.109 µm</td><td>H<sub>2</sub></td><td>0–0 S(6)</td><td>ro-vibrational</td></tr><tr><td>6.910 µm</td><td>H<sub>2</sub></td><td>0–0 S(5)</td><td>ro-vibrational</td></tr><tr><td>6.985 µm</td><td>Ar<sub>II</sub></td><td>2P<sup>3/2</sup>–2P<sup>1/2</sup></td><td>forbidden</td></tr><tr><td>7.460 µm</td><td>H<sub>1</sub></td><td>Pfund α</td><td>recombination</td></tr><tr><td>7.502 µm</td><td>H<sub>1</sub></td><td>Humphreys β</td><td>recombination</td></tr><tr><td>8.025 µm</td><td>H<sub>2</sub></td><td>0–0 S(4)</td><td>ro-vibrational</td></tr><tr><td>8.991 µm</td><td>Ar<sub>III</sub></td><td>3P<sup>2</sup>–3P<sup>1</sup></td><td>forbidden</td></tr><tr><td>9.665 µm</td><td>H<sub>2</sub></td><td>0–0 S(3)</td><td>ro-vibrational</td></tr><tr><td>10.511 µm</td><td>S<sub>IV</sub></td><td>2P<sup>3/2</sup>–2P<sup>1/2</sup></td><td>forbidden</td></tr><tr><td>12.279 µm</td><td>H<sub>2</sub></td><td>0–0 S(2)</td><td>ro-vibrational</td></tr><tr><td>12.369 µm</td><td>H<sub>1</sub></td><td>Humphreys α</td><td>recombination</td></tr><tr><td>12.814 µm</td><td>Ne<sub>II</sub></td><td>2P<sup>3/2</sup>–2P<sup>1/2</sup></td><td>forbidden</td></tr><tr><td>15.555 µm</td><td>Ne<sub>III</sub></td><td>3P<sup>2</sup>–3P<sup>1</sup></td><td>forbidden</td></tr><tr><td>17.035 µm</td><td>H<sub>2</sub></td><td>0–0 S(1)</td><td>ro-vibrational</td></tr><tr><td>18.713 µm</td><td>S<sub>III</sub></td><td>3P<sup>2</sup>–3P<sup>1</sup></td><td>forbidden</td></tr><tr><td>21.829 µm</td><td>Ar<sub>III</sub></td><td>3P<sup>1</sup>–3P<sup>0</sup></td><td>forbidden</td></tr><tr><td>25.890 µm</td><td>O<sub>IV</sub></td><td>2P<sup>3/2</sup>–2P<sup>1/2</sup></td><td>forbidden</td></tr><tr><td>28.219 µm</td><td>H<sub>2</sub></td><td>0–0 S(0)</td><td>ro-vibrational</td></tr><tr><td>33.481 µm</td><td>S<sub>III</sub></td><td>3P<sup>1</sup>–3P<sup>0</sup></td><td>forbidden</td></tr><tr><td>34.815 µm</td><td>Si<sub>II</sub></td><td>2P<sup>3/2</sup>–2P<sup>1/2</sup></td><td>forbidden</td></tr><tr><td>35.349 µm</td><td>Fe<sub>II</sub></td><td>6D<sup>5/2</sup>–6D<sup>7/2</sup></td><td>forbidden</td></tr><tr><td>36.014 µm</td><td>Ne<sub>III</sub></td><td>3P<sup>1</sup>–3P<sup>0</sup></td><td>forbidden</td></tr></tbody></table>

not provide meaningful uncertainty estimates. Galliano et al. (2008b) performed a systematic comparison of the Lorentzian and spline methods, and found that, although band intensities were different, the trends between band intensities or band ratios were consistent with the two methods.

**The template method** consists in fitting a small number of synthetic spectra characteristics of different regions or different species. We have demonstrated this method in Fig. III.34.c using PAH<sub>at</sub> (Pilleri et al., 2012). This method uses four main components. These components were extracted from a sample of Galactic PDRs and PNe, using blind-signal separation methods (Berné et al., 2007; Joblin et al., 2008). In that sense, they are empirical, synthetic small grain spectra. These components are the following.

**Neutral PAHs** (PAH<sup>0</sup>) are shown in Fig. III.35.a. Their brightest C–C band is centered at λ ≈ 7.65 µm.

**Ionized PAHs** (PAH<sup>+</sup>) are shown in Fig. III.35.b. They have similar band centers as PAH<sup>0</sup>, but different intensity ratios, as we have previously seen (cf. Fig. III.32). Both PAH<sup>0</sup> and PAH<sup>+</sup> have similar spectral characteristics as class A spectra of Peeters et al. (2002a), found in H<sub>II</sub> regions, and thought to be “processed” PAHs.

**Large ionized PAHs** (PAH<sup>x</sup>), shown in Fig. III.35.c, have a peculiar redshifted C–C band at λ ≈ 7.9 µm, observed in PNe by Joblin et al. (2008). They are assumed to be ≈ 100 C atom
grains. Those are reminiscent of the class *B* spectra of Peeters et al. (2002a).

**Evaporating VSGs** (eVSGs), shown in Fig. III.35.d, are thought to be PAH clusters of ≈ 500 C atoms. Their spectra have the characteristics of class *C* of Peeters et al. (2002a) seen in post-AGB stars, with a broad 7.9 µm band and no 8.6 µm feature. Peeters et al. (2002a) conjectured these could be “*freshly*-formed” carbon grains. The carrier of the broad 7.9 µm band could be destroyed by the ionizing radiation, during its journey in the ISM (e.g. Joblin et al., 2008). In this scenario, PAHs are formed by the destruction of VSGs at the UV-illuminated edges of molecular clouds. Class *C* is not well studied. Some novae show spectra similar to Class *C*, and could be due to nitrogen impurities in hydrocarbon compounds (Endo et al., 2021).

The advantages of this method are that: (i) fits are not extremely degenerate, even at low signal-to-noise ratios, because of the small number of free parameters, compared to the Lorentzian method; (ii) the four different classes are physically meaningful, providing a clear interpretation of the results. However, its lack of flexibility prevents accurate fits, that could overlook new information present in the observations.

<figure><img src="image_364.png" /></figure>

FIGURE III.35 – *PAH and small grain templates.* We show the four components of PAHtat (Pilleri et al., 2012). Vertical yellow dashed lines indicate the same wavelengths in the four panels, helping visualizing the band shifts of some components. Licensed under CC BY-SA 4.0.

### III.2.1.3 PAH Band Ratio Studies

Applying a spectral decomposition method to a set of MIR spectra allows us to study the variations of several band ratios that contain physical information about the small carbon grain properties.
**Observed band ratios in galaxies.** Fig. III.36 demonstrates the diversity of spectra among galaxies (panel a) or within one (panel b). This figure emphasizes the differences in terms of aromatic band intensity. All these spectra are normalized by the 11.3-µm feature intensity. Yet, they exhibit large variations of their 6-to-9-µm features. This can be more precisely quantified by studying the correlation between specific band ratios, such as in Fig. III.37 (Galliano et al., 2008b). The quantity I(λ) is simply the intensity of the feature centered at λ µm. The bottom two panels show the results for integrated galaxies and Galactic regions, whereas the bottom two panels show the results for a few spatially-resolved sources. Overall the trends are similar for both types. It means that this is a multiscale relation, valid at sub-pc scales (within the Orion bar or M 17) and ≈ 10 kpc scales (among integrated galaxies). These three displayed band ratios span about an order of magnitude, and are linearly correlated with each other. It implies that the 6.2, 7.7 and 8.6 µm features are tied together, while the 11.3 µm can vary independently. This is what was illustrated in Fig. III.36. The only parameter that can explain such a variation is the charge of the PAHs (cf. Fig. III.32).

<figure><img src="image_365.png" /></figure>

The variation of the PAH charge can explain most of the `UIB` variations observed in the nearby Universe.

<figure><img src="image_366.png" /><figcaption>FIGURE III.36 – Diversity of MIR spectra among and within galaxies. Panel (a) shows the integrated spectra of three galaxies: (i) UM 448, a BCD; (ii) M 51, a prototypical LTG; and (iii) NGC 4945, a LIRG. Panel (b) shows the spectra of three different regions within the starbursting irregular galaxy, M 82. In both panels, the monochromatic flux is normalized to its value at 11.3 µm, in order to demonstrate band ratio variations. These are ISO spectra (Galliano et al., 2008b). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Effects of ionization and size.** Although ionization is the main driver of the `UIB` relative variations in galaxies, other effects can play a role in specific environments. The most important one of these secondary effects is the `PAH` size distribution. Fig. III.38 shows numerical simulations of several key `UIB` ratios (Hu et al., 2021a). We have varied: (i) the minimum `PAH` size expressed in number of C atoms, $N_{C}^{min}$, highlighted in Fig. III.38.a; (ii) the `PAH` charge fraction, $f_+$, highlighted in Fig. III.38.b; (iii) the `ISRF` intensity and hardness. To illustrate the last effect, we have computed the model grid for the Solar neighborhood `ISRF` (Mathis et al., 1983). This values are the bright grid points. We have also computed the grid for a hot star spectrum, with $U = 10^4$ (the faint grid points). The effect of the `ISRF` is non negligible, but it is less drastic than that of the charge and size. Most astrophysically relevant `ISRFs` will be intermediate between the two extreme cases we have displayed. Fig. III.38 illustrates
<figure><img src="image_367.png" /></figure>

<figure><img src="image_368.png" /></figure>

FIGURE III.37 – *PAH band ratio correlations inside and among galaxies.* The top two panels show the values of the ratios derived from integrated spectra of galaxies, as well as Galactic and Magellanic regions. The bottom two panels show pixel distribution of the same ratios, within a variety of objects. These data are a combination of ISO/CAM and *Spitzer/IRS* spectra, fitted using the Lorentzian method (Galliano et al., 2008b). The grey-filled curves represent the ±1σ linear fit to the integrated source correlations (cf. Table 2 of Galliano et al., 2008b). The grey curves are identical in panels (a) and (c) and in panels (b) and (d), to guide the eye when comparing integrated and resolved correlations. Licensed under CC BY-SA 4.0.
the following points (see also Rigopoulou et al., 2021, for a more complete calculation using *Density Functional Theory*).

I(3.3)/I(11.3) is mainly a size tracer. Both 3.3 and 11.3 µm bands (C-H modes) are indeed primarily emitted by neutral PAHs. Charge thus does not significantly impact this ratio. The 3.3 µm feature is mainly carried by the smallest PAHs, whereas the 11.3 µm feature is carried by intermediate sizes.

I(7.7)/I(11.3), on the opposite, is essentially a charge tracer. It is still sensitive to the size, because of the relatively large difference in wavelength of both features. The degeneracy due to this additional dependence can be broken, using I(3.3)/I(11.3).

Such band ratio diagrams have been used to demonstrate systematic variations of the PAH size distribution in various environments. We studied the spatial variations of I(3.3)/I(11.3) in NGC 1097 (Fig. III.30) and showed it was systematically lower in the central region, close to the AGN (Wu et al., 2018b). The most likely explanation is that the hard radiation field from the central engine is selectively destroying the smallest PAHs. This effect was also shown by Smith et al. (2007) and Sales et al. (2010), on global scales.

<figure><img src="image_369.png" /></figure>

FIGURE III.38 – *Theoretical MIR band ratio variations.* Both panels show the same grid points, but are highlighted differently. Each circle is the theoretical band ratio estimated from the stochastic emission spectrum of a PAH mixture having the Draine & Li (2007) optical properties and the Zubko et al. (2004, BARE-GR-S) size distribution. The solid color symbol are mixtures heated with the Solar neighborhood ISRF (Fig. I.28) with $U = 1$, whereas the pastel symbols are heated by a black body at $T = 30\,000$ K with $U = 10^4$. In panel (a), we highlight the effect of changing the minimum size cut-off, in number of C atoms, $N_C^{min}$. In panel (b), we show the effect of the charge fraction, $f_+$. Licensed under CC BY-SA 4.0.

**The particular case of low-metallicity environments.** In low-metallicity systems, band ratio variations can be more difficult to probe, as the band equivalent widths are lower, and thus more uncertain (cf. Sect. III.2.1.4). In the LMC, Mori et al. (2012) found different trends in neutral and ionized sightlines. Toward the latter, there are evidences that PAHs have a lower charge (as a consequence of the higher recombination rate) and are on average larger (due to the destruction of the smallest PAHs). In contrast, in the SMC, Sandstrom et al. (2012) found very weak I(6-9)/I(11.3) ratios and weak 8.6 and 17 µm bands, implying small weakly ionized PAHs. This last point is consistent with the trend
* The I(6.2)/I(11.3) has a relatively narrow dynamical range. It varies at most by one order of magnitude in our sample. The relation is unclear outside this range.

Finally, the calibration of Eq. (III.45) can also be estimated theoretically. Fig. III.39.b shows the theoretical I(6.2)/I(11.3) ratio as a function of G₀ and nₑ (Galliano, 2009). It has been derived by computing the stochastic emission of PAHs within the PDR model of Kaufman et al. (2006). Such model indeed computes the charge balance of PAHs at each point within the cloud, where G₀ and nₑ are known. One grid point value corresponds to a whole cloud.

<figure><img src="image_370.png" /></figure>

FIGURE III.39 – *Calibration of PAH ratio diagnostics.* Panel (a) shows the empirical correlation between the observed 6.2-to-11.3 µm ratio and the modeled parameter, γ (Eq. III.44), in three galactic PDRs (Galliano et al., 2008b). Panel (b) shows the theoretical variation of the band ratio as a function of G₀ and nₑ (Galliano, 2009). This parameter grid was computed implementing stochastic heating in the PDR model of Kaufman et al. (2006), using a fixed PAH size distribution (the BARE-GR-S of Zubko et al., 2004). Licensed under CC BY-SA 4.0.

**Other properties.** We end this section by briefly reviewing other properties that can be studied with observations of *UIBs*.

**PAH compactness** can be probed by studying the relative variations of C–H *OOP* bending modes (cf. Fig. III.31.a). These bending modes depend on the number of H atoms per aromatic cycle (cf. Fig. III.31.b). We have seen in Sect. III.2.1.1 that, in particular, the solo-to-trio intensity ratio, I<sub>11.3</sub>/I<sub>12.7</sub>, is an indicator of PAH compactness. This ratio was scrutinized in Galactic regions (evolved stars, HII regions, reflection nebulae and YSOs) by Hony et al. (2001). Their results are consistent with the scenario where large (≈ 100 – 150 C atoms), compact PAHs are formed in winds of evolved stars, and degraded into smaller, irregular molecules in the ISM.

**Small a-C(:H) hydrogenation** can be studied with the I(3.4)/I(3.3) ratio. There is a debate whether aromatic features are carried by PAHs or small a-C(:H). The 3.4 µm aliphatic feature however can not be carried by pure PAHs, it must come from either a-C(:H) grains (Jones et al., 2013) or *Hydrogenated PAHs* (HPAH) with one of several aliphatic groups (e.g. Fig. 1 of Marciniak et al., 2021). The I(3.4)/I(3.3) aliphatic-to-aromatic ratio shows regional variations in the ISM, as the result of structural changes in the hydrocarbons through UV processing (e.g. Jones et al., 2013). Mori et al. (2014) showed that the I(3.4)/I(3.3) decreases with the ionization of PAHs, in Galactic HII regions. Yamagishi et al. (2012) detected the 3.4 µm feature in the superwind of M 82.

<figure><img src="image_371.png" /></figure>
<figure><img src="image_372.png" /><figcaption>FIGURE III.40 – PAH band ratio as diagnostics of the physical conditions. These are the data shown in Fig. III.37 where the I(6.2)/I(11.3) ratio as been converted to G₀/nₑ × √T using Eq. (III.45) (Eq. (5) of Galliano et al., 2008b). Licensed under CC BY-SA 4.0.</figcaption></figure>

They found that the I(3.4)/I(3.3) ratio increases with distance from the center. They interpreted this trend as the production of small *a-C(:H)*, by shattering of larger grains in this harsh halo. Similarly, Kondo et al. (2012) found a higher I(3.4)/I(3.3) ratio in the nuclear bar of NGC 1097, suggesting that the gas flow towards the center could lead to the formation of small *a-C(:H)* by shattering. We note that, alternatively, the I(3.4)/I(3.3) ratio can increase with the accretion of *a-C(:H)* mantles in denser regions (Jones et al., 2013). This feature can also be seen in extinction, in AGNs (e.g. Mason et al., 2007). Hu et al. (in prep.) modeled the spatially resolved AKARI and Spitzer spectra in M 82. In this study, we found a negative correlation between the I(3.4)/I(3.3) and [S IV]₁₀.₅₁µm/[Ne II]₁₂.₈₁µm ratios. The latter is a ISRF hardness indicator. This result thus demonstrates the dehydrogenation of *a-C(:H)* grains by hard ISRFs.

**SFR indicators** are one of the most sought after astrophysical diagnostics. It happens that *UIBs* can be used to that purpose. Peeters et al. (2004) showed that the 6.2 µm feature intensity correlates well with *SFR*, making it a reliable estimator. Alternatively, Shipley et al. (2016) have calibrated a *SFR* estimator based on the integrated power of the 6.2, 7.7 and 11.3 µm features. The reason of this correlation is the same as for the *TIR-SFR* correlation (e.g. Kennicutt & Evans, 2012, for a review). At first order, the *UIB* strength is indeed correlated with *TIR*. Peeters et al. (2004) however note that *UIBs* are biased towards B stars. In addition, we note that such a tracer will be underestimating the *SFR* at low metallicity, because of the variation of the relative *UIB* strength, as we will see in Sect. III.2.1.4.

### III.2.1.4 Variations of the Aromatic Feature Strength

The evolution of the shape of the *UIB* spectrum, probed by studying band ratio variations, is not the only diagnostics of the small carbon grain properties. The overall aromatic feature strength, relative to the continuum (i.e. to the emission of the rest of the dust populations) shows drastic variations across environments (cf. Fig. III.30). These variations trace the evolution of the mass fraction of their carriers – *PAH* or small *a-C(:H)*.
**Effect of ISRF hardness.** PAH and small a-C(:H) are known to be sensitive to hard an intense radiation fields. They tend to evaporate near massive stars, and can be assumed to be fully depleted in H II regions (e.g. Cesarsky et al., 1996b; Galametz et al., 2013; Galliano et al., 2018). This effect can be quantified by studying the variation of the aromatic feature strength with a tracer of the ISRF hardness.

1. The aromatic feature strength can be traced with I(PAH)/I(cont), the PAH-to-MIR-continuum ratio, where I(PAH) is the sum of the intensities of every aromatic feature, and I(cont) is the integrated intensity of the continuum, over an arbitrary wavelength range (10-16 µm in the case of Madden et al., 2006).

2. The hardness of the ISRF can conveniently be traced by the [Ne III]15.56µm/[Ne II]12.81µm ratio, as both of these MIR lines are usually bright. Madden et al. (2006) demonstrated, using a photoionization model, that [Ne III]15.56µm/[Ne II]12.81µm is around unity when the ISM is heated by a young star cluster, and drops rapidly after a few million years, as massive stars die.

Such a trend is shown on Fig. III.41, with the Madden et al. (2006) results<fnref n="8" />. It clearly indicates that PAHs are less abundant in regions permeated with a hard ISRF, at all spatial scales down to ≈ 0.1 pc. Note however that both tracers do not come from the same physical region: (i) PAH being destroyed in H II regions are tracing the neutral gas; (ii) [Ne III]15.56µm and [Ne II]12.81µm obviously come from the ionized gas. The correlation therefore reflects the mixing of phases within the beam.

<figure><img src="image_373.png" /></figure>

FIGURE III.41 – *Effect of ISRF hardness on PAH strength.* These results are from the spectral decomposition of Madden et al. (2006) and Galliano et al. (2008b). Panel (a) shows integrated sources, whereas panel (b) shows pixel-by-pixel distributions. The white line with the grey band shows the affine fit to the global values, ±1σ, in log-log space. It is the same in both panels. Licensed under CC BY-SA 4.0.

**Effect of metallicity.** The effects of ISRF and metallicity are often degenerate, as at low metallicity: (i) stars of a given mass have a systematically higher effective temperature, because of line blanking effects; (ii) the ISM is less opaque, because of the lower dustiness, and thus more permeated by UV radiation. The highest [Ne III]15.56µm/[Ne II]12.81µm ratios are found in BCDs, as well as the lowest

<footnote n="8">8. The results of Madden et al. (2006) consisted in a first spectral decomposition of ISO/CAM spectra, that we have refined in Galliano et al. (2008b), and added Spitzer/IRS spectra to the sample.</footnote>
I(PAH)/I(cont). Fig. III.42.a shows the evolution of I(PAH)/I(cont) as a function of metallicity (Madden et al., 2006). There is a paucity of PAHs in low-metallicity environments. The questions is whether this is the result of their increased destruction, or if they have not been produced. We will come back to this question, when discussing dust evolution in Sect. IV.2.2.1.

<figure><img src="image_374.png" /></figure>

PAHs are under-abundant in low-metallicity environments.

**The absence of metallicity threshold.** The relation of Madden et al. (2006) was the first spectroscopy-based demonstration of the effect. Shortly before, Engelbracht et al. (2005) showed the broad-band correlation of $F_v(8~\mu m)/F_v(24~\mu m)$ as a function of metallicity. They showed both quantities were clearly correlated. They however argued there were essentially two populations, below and above $12 + \log(O/H) \approx 8$. Galliano et al. (2008a) showed that this was a bias due to the saturation of $IRAC_{8\mu m}$ as a PAH tracer at low metallicity. When the aromatic feature strength becomes indeed low, $F_v(8~\mu m)/F_v(24~\mu m)$ is not anymore a measure of I(PAH)/I(cont), but is a measure of the temperature of the continuum. This is illustrated on Fig. III.42.b. We show that when the actual mass fraction of small a-C(:H) drops below $\approx 10\%$, $F_v(8~\mu m)/F_v(24~\mu m)$ becomes insensitive to its value.

<figure><img src="image_375.png" /></figure>

FIGURE III.42 – *Effect of metallicity on the PAH strength.* Panel (a) shows the trend of PAH relative strength with metallicity, by Madden et al. (2006). This was the first spectroscopy-based demonstration of this relation. Panel (b) illustrates the bias of the $F_v(8~\mu m)/F_v(24~\mu m)$ ratio as a tracer of PAH strength. It shows the flux ratio as a function of the modeled fraction of small a-C(:H) (the grains carrying the aromatic features), $q_{AF}$, in the DustPedia sample (G21). Only galaxies with both $IRAC_{8\mu m}$ and $MIPS_{24\mu m}$ fluxes are displayed. Licensed under CC BY-SA 4.0.

### III.2.2 Long-Wavelength Properties

At long wavelengths, in the submm-to-cm range, dust emission does not always behave as the extrapolation from the Rayleigh-Jeans law ($F_v \propto v^{2+\beta}$). Two peculiar phenomena have been widely discussed in the literature: (i) the *submillimeter excess*; and (ii) the *Anomalous Microwave Emission (AME)*.
### III.2.2.1 The Elusive Submillimeter Excess

An excess emission above the modeled dust continuum is often observed, longward $\lambda \approx 500$ µm. The most significant reports of this submm excess can not be accounted for by: (i) thermal dust emission; (ii) free-free and synchrotron continua; and (iii) molecular line emission (cf. Fig. III.43; Galliano et al., 2003).

1. The first occurrence of such an excess was unveiled by Reach et al. (1995), studying the COBE observations of the Milky Way. Their IR-submm SED could be fitted with a MBB ($\beta = 2$; cf. Sect. III.1.2.1), and an additional 4–7 K component.

2. A few years later, Lisenfeld et al. (2002) and Galliano et al. (2003) found a statistically significant excess in the dwarf galaxy NGC 1569, at 850 µm and 1.3 mm. It is shown in Fig. II.10.

3. Several subsequent studies confirmed the presence of an excess in other late-type galaxies (e.g. Dumke et al., 2004; Bendo et al., 2006; Zhu et al., 2009; Galametz et al., 2009, 2011), including the global SEDs of the Magellanic clouds (Israel et al., 2010; Bot et al., 2010).

4. Herschel and Planck opened the way to more detailed characterizations.

Studying this excess is important, as: (i) it could bias dust mass estimates; (ii) it potentially contains untapped physical information about the ISM.

<figure><img src="image_376.png" /></figure>

**FIGURE III.43 – Submillimeter excess in NGC 1569.** This is the best fit model of Galliano et al. (2003), discussed in Sect. III.1.3.4. We focus here on the blue component, which is a T = 5 K, $\beta = 1$ MBB accounting for ≈ 80% of the dust mass of the galaxy. This component has been fitted to account for the submm excess. Licensed under CC BY-SA 4.0.

**Possible explanations.** The origin of the submm excess is currently debated. The following explanations have been proposed. These different scenarios are not exclusive.

**Very cold dust (VCD),** that is equilibrium grains colder than T ≈ 10 K, can be used to fit the excess. The emission of such a grain population indeed peaks in the submm range. Cold dust being weakly emissive, this scenario however leads to massive amounts of grains (40–80% of the total dust mass; e.g. Galliano et al., 2003, 2005). Galliano et al. (2003) showed that VCD would be realistic only if this component was distributed in a few number of dense, parsec-size clumps. The existence of such cold dust indeed requires it to be shielded from all UV-visible light. Using the spatially resolved observations of the 500 µm excess in the LMC, Galliano et al. (2011)
concluded that this explanation is unrealistic, as it would require at least one clump in each of
the ≈ 90 000 pixels of this study.

**Temperature-dependent emissivity.** Laboratory measures on cosmic grain analogs exhibit an increase of the FIR-submm opacity as a function of temperature (e.g. Mennella et al., 1998; Demyk et al., 2017a,b). Meny et al. (2007) designed the so-called *Disordered Charge Distribution/Two-Level System* (DCD/TLS) model, predicting an increase of $κ(λ_0)$ and a decrease of $β$ with the temperature of amorphous grains. It reproduces the submm excess observed in the Galaxy by Paradis et al. (2012) and in the LMC by Bot et al. (2010), coupled with spinning grains; cf. Sect. III.2.2.2). However, this model can not account for the submm excess in the SMC (Bot et al., 2010).

**Magnetic grains.** Draine & Hensley (2012) showed that the submm excess of the SMC could be attributed to magnetic nanoparticles (Fe, $Fe_3O_4$, γ-$Fe_2O_3$ $^9$). Thermal fluctuations in the magnetization of these grains can produce strong magnetic dipole emission, since ferromagnetic materials are known to have large opacities at microwave frequencies. This hypothesis seems to be consistent with the observed elemental abundances of the SMC and could also be responsible for the excess detected in other environments. These magnetic nanoparticles could be free-flying or inclusions in larger grains.

**Empirical properties of the excess.** Since the origin of the submm excess is still unknown, most studies focus on characterizing its observed properties.

**Low-metallicity systems** are the environments where the excess appears to be the most prominent. This is the reason why it has been detected in most nearby BCDs (e.g. Galliano et al., 2003; Dumke et al., 2004; Galliano et al., 2005; Galametz et al., 2009; Bot et al., 2010; Galametz et al., 2011). Galliano et al. (2011) have shown, using spatially-resolved observations of the LMC, that the SPIRE$_{500µm}$ excess, $r_{500}$, varies up to ≈ 40% in certain regions. It is correlated with the mean starlight intensity, $\langle U \rangle$, and anticorrelated with the dust surface density, as shown in Fig. III.44. The correlation with $\langle U \rangle$ could be a confirmation of the DCD/TLS effect, as $\langle U \rangle$ directly controls the temperature of the grains. The correlation with surface density is however significantly better than with ISRF, a trend that the DCD/TLS model alone fails to explain.

**In the Milky Way, Paradis et al. (2012) showed that the SPIRE$_{500µm}$ excess becomes significant in the peripheral regions ($> 35°$), as well as towards some H II regions. This is qualitatively consistent with the trend we found in the LMC, as these peripheral regions are also the most diffuse ones. The relative amplitude of the excess can rise up to ≈ 20%**.

**In other galaxies,** the excess is more difficult to detect (e.g. Rémy-Ruyer et al., 2015; Dale et al., 2017). When resolved in non-barred spirals, the submm excess is primarily detected in the disk outskirts, thus at low-surface density (e.g. Hunt et al., 2015).

<figure><img src="image_377.png" /></figure>

The submm excess is more prominent in low-metallicity environments, and in diffuse regions.

**Reality of the phenomenon.** The reality of the submm excess has been questioned for the two following reasons. We try to address these criticisms in order to support its likeliness.

**The excess is model-dependent.** Different dust opacities lead to different amplitudes of the excess. Some over-parametrized models have found excesses virtually everywhere. For instance, the *Broken-Emissivity Modified Black Body* (BEMBB) is a MBB with two free-varying emissivity indices, $β_1$ and $β_2$, below and above $λ = λ_b$ (Gordon et al., 2014). With such a model, any excess

<footnote n="9">9. γ-Fe₂O₃ is the notation to design ferromagnetic Fe₂O₃, called *maghemite*. It must be distinguished from its non-magnetic form, noted α-Fe₂O₃, called *hematite*. Both have distinct crystalline structures: cubic lattice for maghemite; trigonal crystals for hematite (e.g. Łęcznar, 1977).</footnote>
<figure><img src="image_378.png" /></figure>

<figure><img src="image_379.png" /><figcaption>FIGURE III.44 – *Spatially-resolved submm excess in the LMC.* In both panels, the color density repre-
sents the number of pixels per bin of parameters in the star-forming region N44 of the LMC (Galliano
et al., 2011). The SPIRE$_{500µm}$ excess is shown as a function of: (a) mean starlight intensity (Eq. III.38);
and (b) dust mass surface density. Licensed under CC BY-SA 4.0.</figcaption></figure>

can be fitted, but the derived β₁, β₂ and λb do not necessarily correspond to existing solids.
This model is also extremely degenerate, as shown by Galliano (2018). A BEMBB fit of the dif-
fuse Galactic ISM SED (cf. Fig. II.25.d) by Gordon et al. (2014) gives a r<sub>500</sub> ≈ 48 ± 11% excess
starting at λb ≈ 300 ± 30 µm. Yet, this is the SED that everyone uses to calibrate dust models.
It does not exhibit an actual excess. It illustrates that probing the submm excess with models
which are not based on realistic optical properties is a non-sense. It must be studied with a
model as physical as possible.

**The technically-challenging nature of submm-mm observations** is also questioning the reality of the excess: (i) observations in this regime are difficult from the ground (cf. Sect. II.1.1.1); (ii) the calibration of these observations is often uncertain; (iii) data reduction methods have problems dealing with the diffuse emission, which is where the excess appears to be the most prominent. In particular, *Planck* data have lead to revise the calibration of the COBE/FIRAS FIR-submm spectrum of the diffuse Galactic ISM (Planck Collaboration et al., 2014c). With this new calibration, the excess of Reach et al. (1995), discussed earlier, is not significant anymore. There are however several sound indications that these concerns are not enough to doubt the reality of the phenomenon.

* The SED shape of low-metallicity systems is well characterized in this regime. It has been observed at different wavelengths, with different instruments. It is still present with the latest *Herschel* calibration (Dale et al., 2017).

* Reports of a deficit are very rare.

* In the Magellanic clouds, Planck Collaboration et al. (2011a) showed that, while the submm excess in the integrated SED of the LMC was consistent with CMB fluctuations, the SMC excess was significantly above this level.

III.2.2.2 The Anomalous Microwave Emission

As we have seen in *Sect. II.2.2.3*, the *AME* is a centimeter continuum excess that can not be accounted for by: (i) the extrapolation of dust thermal emission; (ii) molecular line emission; and (iii) free-free
<figure><img src="image_380.png" /></figure>

and synchrotron continua (Fig. II.10). Its SED looks like a bump peaking around $\lambda = 1$ cm (cf. Fig. II.10). It is commonly attributed to the dipole emission of fastly rotating, small dust grains. The faster grains rotate, the shorter the frequency of the emission peak is.

**The AME in extragalactic environments.** In nearby galaxies, the first unambiguous detection of an AME has been obtained in an outer region of NGC 6946 (Murphy et al., 2010; Scaife et al., 2010). Follow up observations showed evidence for AME in eight regions of this galaxy (Hensley et al., 2015). This study showed that the spectral shape of this AME is consistent with spinning dust, but with a stronger AME-to-PAH-surface-density ratio, hinting that other grains could be the carriers. Overall, the AME fraction is highly variable, in nearby galaxies. Peel et al. (2011) put upper limits on the AME in M 82, NGC 253 and NGC 4945. These upper limits suggest that AME/100 µm is lower than in the Milky Way, in these objects. In M 31, Planck Collaboration et al. (2015a) report a 2.3σ measurement of the AME, consistent with the Galactic properties. Finally, Bot et al. (2010), fitting the NIR-to-radio SED of the LMC and SMC, tentatively explained the submm/mm excess with the help of spinning dust, in combination with a modified submm dust emissivity (cf. Sect. III.2.2.1). They concluded that, if spinning grains are responsible for this excess, their emission must peak at 139 GHz (LMC) and 160 GHz (SMC), implying large ISRF intensities and densities. Draine & Hensley (2012) argued that such fastly rotating grains would need a PDR phase with a total luminosity more than two orders of magnitude brighter than the SMC.

**Controversy about the carriers of the AME.** Although PAHs have been considered the most likely carriers of the AME, Hensley & Draine (2017) argued that nanosilicates could be a viable alternative. They showed that nanosilicates can indeed account for the AME, without violating the other observational constraints (depletions, emission, extinction; cf. Sect. II.2). This claim relies on the earlier findings of Hensley et al. (2016), showing that AME correlates better with dust radiance, $\mathcal{R}^{10}$, in the MW. Hensley et al. (2016) also found some fluctuations in the PAH emission relative to the AME intensity, traced by WISE$_{12\mu m}$. The fact is that there is no observational evidence of nanosilicates in the ISM. In particular, these grains would emit the 9.8 and 18 µm features, as they would be stochastically heated. These bands would eventually remain diluted in the aromatic feature emission, provided the abundance of nanosilicates is low enough. We can however note that in HII regions, where PAHs are destroyed, we can see the underlying continuum, and silicate features in emission are rarely present (e.g. Peeters et al., 2002b). They can be seen only in the most compact HII regions, and their 9.8-to-18 µm ratio indicates that they originate in large equilibrium grains. A weak correlation does not always indicate an absence of causality. This issue might reside in the fact that Hensley et al. (2016) used the WISE$_{12\mu m}$ band as a tracer of PAH intensity, whereas this broad photometric band also encompasses a significant fraction of the underlying continuum, and is biased towards neutral PAHs. We have tried to address this controversy by refining the constraints on the PAH emission.

**Correlation with charged PAHs.** Bell et al. (2019) focussed on the 10°-wide Galactic molecular ring surrounding the O-type star, $\lambda$-Orionis (cf. Fig. III.45). We chose this region, because the Planck data indicate there is a large gradient of AME intensity. We fitted the spatially-resolved SED, at 0.25° angular sampling, using the AKARI 9 µm and IRAS 12 µm bands to constrain the PAH abundance, and longer wavelength bands for the rest of the emission. We used the dust SED code HerBIE (HiERarchical Bayesian Inference for dust Emission; Galliano, 2018, cf. Sect. V.3.3). We were able to address the controversy, thanks to the combination of: (i) rigorous SED fitting, allowing us to account for all the available information, not only a few broadbands; (ii) better observational constraint on the PAH emission (9 and 12 µm); and (iii) focussing on a clean region rather than the whole sky. Our results are shown in Fig. III.46. We have found very good pixel-to-pixel correlations between the AME intensity, derived by Planck Collaboration et al. (2016a), and the dust and PAHs surface densities from

<figure><img src="image_381.png" /></figure>

<footnote n="10">10. The radiance is the spectral integral of the specific intensity: $\mathcal{R} \equiv \int I_v dv$.</footnote>
SED fitting (e.g. cf. Fig. III.46.a). Our analysis however show that, if the dust mass per pixel is very well correlated with the intensity of AME per U, $I_{AME}/U$ ($\rho \approx 0.88$), the correlation is better with the mass of PAHs, and even better with the mass of ionized PAHs ($\rho \approx 0.92$; cf. Fig. III.46.b). Our Bayesian results also show that there is a 0 probability that the total dust could correlate better with AME than with PAHs. Our impression is that the study of Hensley et al. (2016) may have too quickly interpreted a poor correlation as an absence of causality. The scatter in WISE$_{12\mu m}$ was likely not the result of an intrinsic scatter of the PAH/AME correlation, but rather a combination of observational artefacts: (i) noise; (ii) contribution of the continuum and ionic lines. Our SED model and our better MIR coverage allowed us to more accurately recover the true PAH column density.

☞ AME correlates better with charged PAHs.

<figure><img src="image_382.png" /><figcaption>FIGURE III.45 – AKARI 9 µm map of λ-Orionis. This is a mosaic of broadband photometric images of PAHs within this Galactic molecular ring. Credit: Bell et al. (2019).</figcaption></figure>
<figure><img src="image_383.png" /></figure>

FIGURE III.46 – AME correlation with ionized PAHs in λ -Orionis. Panel (a) shows the pixel correlation between I<sub>AME</sub>/U and M<sub>PAH+</sub>, derived from the SED modeling of (Bell et al., 2019). Each point and its uncertainty ellipse represent the posterior PDF of a pixel. Panel (b) shows the posterior PDFs of the correlation coefficients between two sets of parameters (such as panel a). The correlation between I<sub>AME</sub>/U and M<sub>PAH+</sub> is the best of our study, and it is significantly better than with M<sub>dust</sub>. Licensed under CC BY-SA 4.0.

III.3 Dust in Relation with the Gaseous and Stellar Contents

We end this chapter with a short introduction to *ISMology*, since the knowledge of the *ISM* as a whole
is crucial to understanding *ISD*. We discuss a few of our results in this domain and illustrate the way
dust can be used to better understand the gas. The video lectures and accompanying slides of the
2021 “ISM of galaxies” summer school (35 hours of lecture in total), that we have organized, can
provide a good introduction to this subject. Otherwise, the books of *Spitzer* (1978), *Tielens* (2005)
and *Draine* (2011) are references.

III.3.1 The Phases of the ISM

The ISM is intrinsically a *multi-phase* medium, with large contrast densities and differences in tem-
peratures. The order of magnitude of its average density is n<sub>gas</sub> ≈ 1 cm<sup>-3</sup>. We list the physical char-
acteristics of the main phases in *Table III.6*. We will discuss each phase individually in the rest of this
section.

**The cooling function.** The way the gas cools across phases has a decisive impact on the multiphase structure of the ISM. It is possible to estimate its cooling rate as a function of temperature, or, in other words, how the thermal energy of the gas is converted into radiation. This quantity is called the *cooling function*. It is represented on Fig. III.47. We have highlighted the dominant processes in the different temperature regimes.

* At low-temperatures, in neutral atomic regions, [CII]158µm is the main coolant.

* Around T<sub>gas</sub> ≈ 10<sup>4</sup> K, there is drastic increase of cooling efficiency, thanks to the Lyα<sub>121.6nm</sub> line of H.

* Above T<sub>gas</sub> ≳ 10<sup>4</sup> K, the gas becomes rapidly ionized. The cooling is then channelled through various ions.
<figure><img src="image_384.png" /></figure>

<table><thead><tr><th></th><th>Density</th><th>Temperature</th><th>Volume filling factor</th></tr></thead><tbody><tr><td>HIM</td><td>≈ 3 × 10<sup>-3</sup> cm<sup>-3</sup></td><td>≈ 10<sup>6</sup> K</td><td>≈ 50%</td></tr><tr><td>H II regions</td><td>≈ 1 – 10<sup>5</sup> cm<sup>-3</sup></td><td>≈ 10<sup>4</sup> K</td><td>≈ 1%</td></tr><tr><td>WIM</td><td>≈ 0.1 cm<sup>-3</sup></td><td>≈ 10<sup>4</sup> K</td><td>≈ 25%</td></tr><tr><td>WNM</td><td>≈ 0.3 cm<sup>-3</sup></td><td>≈ 10<sup>4</sup> K</td><td>≈ 30%</td></tr><tr><td>CNM</td><td>≈ 30 cm<sup>-3</sup></td><td>≈ 100 K</td><td>≈ 1%</td></tr><tr><td>Diffuse H<sub>2</sub></td><td>≈ 100 cm<sup>-3</sup></td><td>≈ 50 K</td><td>≈ 0.1%</td></tr><tr><td>Dense H<sub>2</sub></td><td>≈ 10<sup>3</sup> – 10<sup>6</sup> cm<sup>-3</sup></td><td>≈ 10 K</td><td>≈ 0.01%</td></tr></tbody></table>

TABLE III.6 – *Phases of the ISM.* Adapted from Tielens (2005) and Draine (2011). The sum of the filling factors is slightly larger than 100%, because these estimates are rough.

* At $T_{gas} \gtrsim 10^7$ K, in coronal phases, the gas cools mainly *via* free-free emission.

The exact shape of this cooling function can vary sensibly. It depends on: (i) metallicity, as this parameter impacts the relative abundances of the different species; and (ii) the radiation field, which impacts the ionization state of the gas.

### III.3.1.1 The Neutral Atomic Gas

The neutral atomic gas is the most abundant phase in the MW: it accounts for ≈ 60% of the total ISM mass, and ≈ 8% of the total baryonic mass (stars and ISM). It fills up about ≈ 30% of the MW volume. The neutral gas is far from thermal equilibrium, but it is roughly in pressure equilibrium, with:

$$ \frac{P_{\text{gas}}}{k} = n_{\text{gas}} \times T_{\text{gas}} \approx 3000 \text{ K/cm}^3. \qquad (\text{III.46}) $$

**The photoelectric heating.** In neutral regions, the direct heating of the gas by absorption of stellar UV photons is not efficient, because only a small fraction of these photons can be absorbed through the different available atomic lines. In these regions, the heating of the gas is indirect. Dust grains absorb much more efficiently UV photons, with their spectrally continuous cross-section. The absorption of an energetic photon (of a few eV) can lead to the ejection of an electron, *via* the photoelectric effect. This electron will then collide with the gas and heat it. This is the *photoelectric heating* of the gas (de Jong, 1977; Draine, 1978; Bakes & Tielens, 1994; Weingartner & Draine, 2001b; Kimura, 2016). We have schematically represented this process in Fig. III.48. The overall efficiency of this process is related to the integrated surface of dust grains. According to Table II.3, this surface is dominated by small grains. The smallest grains, especially PAHs, are therefore responsible for most of this heating. Wolfire et al. (1995) give an expression for the photoelectric heating rate:

$$ n_{\text{gas}}\Gamma \approx \epsilon_{\text{PE}}(\gamma, T_{\text{gas}}) \times G_0 \times \frac{n_{\text{gas}}}{1 \text{ cm}^{-3}} \times 10^{-25} \text{ W/m}^{-3}, \qquad (\text{III.47}) $$

where $\epsilon_{PE}$ is the efficiency of conversion of FUV energy into gas heating (it is a few percents), and $G_0$ and $\gamma$ have been defined in Eqs. (III.43) – (III.44).

**The two stable neutral atomic phases.** For simplicity, let's assume the heating of the neutral ISM is assured only by photoelectric heating. Let's also use Eq. (III.47) with a fixed value of the photoelectric heating efficiency, $\epsilon_{PE} = 4.9\%$, and $G_0 = 1.7$. The equilibrium is obtained when cooling and heating are balanced:

$$ n_{\text{gas}}\Gamma = n_{\text{gas}}^2 \Lambda(T), \qquad (\text{III.48}) $$

which becomes, using Eq. (III.46):

$$ P_{\text{gas}}/k = \frac{\Gamma \times T}{\Lambda(T)}. \qquad (\text{III.49}) $$

<figure><img src="image_385.png" /></figure>
<figure><img src="image_386.png" /><figcaption>FIGURE III.47 – Cooling function of the ISM at Solar metallicity. The data below T$_{gas}$ = 10$^4$ K is from Dalgarno & McCray (1972), and above, from Schure et al. (2009). We have highlighted the main cooling elements. In the neutral regime, we have assumed an electron fraction of x ≡ n$_{e}$/n$_{H}$ = 10$^{-4}$. Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_387.png" /></figure>

<figure><img src="image_388.png" /><figcaption>FIGURE III.48 – *Photoelectric heating in PDRs.* We have implicitly assumed that the stellar power, $L_*$, is also the total power absorbed by dust, as PDRs are mostly optically thick at UV wavelengths. Licensed under CC BY-SA 4.0.</figcaption></figure>

This is the black line we have represented in Fig. III.49. There are three pressure equilibrium positions (the three dots), at the value of Eq. (III.46). We have hatched in grey the regime corresponding to unstable solutions. In this regime, the pressure indeed decreases with density. Thus, a small pressure increase, above the green dot, will decrease the density. It will thus make the gas expand and its temperature increase, moving further away from the green dot. On the opposite, a small decrease of the pressure, below the green dot, will increase the gas density, and make its temperature decrease, at the same time. The gas will thus collapse and move further away from the green dot. The two only stable solutions correspond to the two main neutral atomic phases of the ISM.

**The Warm Neutral Medium (WNM)** is a diffuse phase with density of the order of $n_{gas} \approx 0.3 \text{ cm}^{-3}$ and temperature $T_{gas} \approx 10^{4} \text{ K}$. It is heated essentially by the photoelectric effect, and also by cosmic rays. It cools via UV-optical emission lines, essentially $Ly\alpha_{121.6\text{nm}}$.

**The Cold Neutral Medium (CNM)** is a denser phase with density of the order of $n_{gas} \approx 30 \text{ cm}^{-3}$ and temperature $T_{gas} \approx 100 \text{ K}$. It is heated essentially by the photoelectric effect, and also by cosmic rays. It cools essentially *via* fine-structure lines, $[CII]_{158\mu m}$ and also $[OI]_{63\mu m}$.

Both of these phases are observed with $[H\text{I}]_{21\text{cm}}$, and UV-optical absorption lines.

<figure><img src="image_389.png" /></figure>

There are two distinct neutral atomic phases in pressure equilibrium: the WNM and the CNM.

### III.3.1.2 The Ionized Gas

The ionized gas accounts for a moderate mass fraction of the ISM in the MW, but fills up a large volume. It is ≈ 23% of the ISM mass and ≈ 3% of the total baryonic mass (stars and ISM).

**The Hot Ionized Medium (HIM).** The HIM is a coronal gas ($n_{gas} \approx 3 \times 10^{-3} \text{ cm}^{-3}$, $T_{gas} \approx 10^{6} \text{ K}$; Table III.6) filling up ≈ 50% of the MW volume. It is at pressure equilibrium with the WNM and CNM. McKee & Ostriker (1977) showed that this phase was heated by SN shockwaves. It is composed of overlapping SN bubbles, and is sometimes referred to as the *intercloud medium*. This gas, which pervades everywhere, can be found significantly above the Galactic disk, contrary to the other phase, which are contained to the disk. It is observed through FUV absorption lines of ions, and diffuse soft X-ray and synchrotron emissions.

**HII regions.** HII regions are short-lived dense ionized regions around OB star associations. The gas is ionized by photons from the massive stars. These regions are not in equilibrium, they are
<figure><img src="image_390.png" /></figure>

<figure><img src="image_391.png" /><figcaption>FIGURE III.49 – Neutral ISM phase diagram. The black line is Eq. (III.49), with the cooling function of Fig. III.47. The horizontal magenta line is the typical pressure of the ISM (Eq. III.46). There are three pressure equilibrium solutions (the three dots), but one (the green dot) lies in the unstable regime (grey hatched). The arrows along each curve, near the dots, show the direction the gas would move if there was a perturbation around the equilibrium. Licensed under CC BY-SA 4.0.</figcaption></figure>

expanding. Strömgren (1939) has estimated the radius of an homogeneous sphere of ionized gas, around a star of H-ionizing photon rate, Q₀. Such a sphere is called a *Strömgren sphere*. Its radius, Rₛ, can be estimated by balancing photoionization and recombination:

$$ \underbrace{Q_0}_{\text{total ionization rate}} = \underbrace{\frac{4\pi}{3} R_s^3 n_p}_{\text{number of protons}} \times \underbrace{n_e \alpha_B}_{\text{recombination rate}}, \qquad (\text{III.50}) $$

where αₛ is the *case B recombination rate*. The product nₑαₛ is the electronic recombination rate to any H level n ≥ 2:

$$ \alpha_B(T_e) \approx 2.6 \times 10^{-13} T_e^{-3/4} \text{cm}^3/\text{s}. \qquad (\text{III.51}) $$

Recombination down to n = 1 will indeed produce an ionizing photon that will be absorbed by another H atom. Rearranging Eq. (III.50), Strömgren's radius is thus:

$$ R_s = \left( \frac{3Q_0}{4\pi n_{\text{gas}}^2 \alpha_B} \right)^{1/3}. \qquad (\text{III.52}) $$

For an O5 star, with nᵣₐₗ = 10³ cm⁻³, Rₛ ≈ 1 pc. This estimate can be refined, accounting for He and other atomic species, as well as dust screening (e.g. Osterbrock, 1989, for an extensive lecture).

<figure><img src="image_392.png" /></figure>
**The Warm Ionized Medium (WIM).** The WIM is a diffuse phase ($n_{gas} \approx 0.1 \text{ cm}^{-3}$, $T_{gas} \approx 10^4 \text{ K}$; Table III.6) filling up about $\approx 25\%$ of the MW volume. It is roughly in pressure equilibrium with the other thermal phases, although it can be found expanding in some regions. This gas is photoionized by photons from OB star associations, escaping from H II regions. The electrons ejected by this photoionization provide the main heating source. It is observed through optical lines, essentially $H\alpha_{656.3\text{nm}}$, as well as some fine-structure lines and free-free emission.

### III.3.1.3 The Molecular Gas

The molecular gas constitutes only a moderate fraction of the ISM mass and a small fraction of its volume. It is however crucial, as it is where stars are formed. In the MW, $\approx 17\%$ of the ISM mass is molecular, which corresponds to $\approx 2\%$ of the total baryonic mass (stars and ISM).

**Molecular hydrogen formation.** The formation of the most abundant molecule of the Universe, H$_2$, is inefficient in the gas phase. This is because the freshly formed molecule needs to evacuate 4.5 eV to remain stable. Yet, due to its symmetry, H$_2$ does not have rotational transitions that would allow it to radiate at these energy levels. H$_2$ formation thus takes place on grain surfaces (e.g. Bron et al., 2014, for a review). We have represented the two main processes on Fig. III.50: the so-called *Langmuir-Hinshelwood* (physisorption) and *Eley-Rideal* (chemisorption) mechanisms. In both cases, the grain serves as a catalyst and helps evacuate the formation energy when the newly formed molecule is released in the gas phase. Similarly to the photoelectric heating, this process happening on grain surfaces, it takes place primarily on small grains. Temperature fluctuations therefore play an important role in its efficiency (e.g. Le Bourlot et al., 2012).

<figure><img src="image_393.png" /><figcaption>(b) ELEY-RIDEAL</figcaption></figure>

<figure><img src="image_394.png" /></figure>

FIGURE III.50 – *H$_2$ formation on grain surface.* Panel (a) illustrates the *Langmuir-Hinshelwood mechanism*, which is driven by physisorption. H atoms are first captured by the grains and migrate to form H$_2$. The energy released by the formation liberates the molecule. Panel (b) demonstrates the *Eley-Rideal mechanism*, which is driven by chemisorption. In this case, an H atom is absorbed at a site where another H atom is already present. See Bron (2014) for more details. Licensed under CC BY-SA 4.0.

**The diffuse molecular gas.** Molecular gas can be observed at moderate densities ($n_{gas} \approx 100 \text{ cm}^{-3}$; Table III.6). This is often associated with the CNM with large enough column densities to allow H$_2$ to be self-shielded (i.e. its UV electronic lines are optically thick). It is also heated by photoelectric effect and cosmic rays. It cools primarily *via* $[CII]_{158\mu m}$ and can be observed through UV absorption lines.

**Photodissociation regions.** Same as H II regions, PDRs are a phase defined by the presence of massive stars in their vicinity. They are not a stable phase of the ISM, but they are very important since
<figure><img src="image_395.png" /></figure>

they are the interface between the HII region and the molecular cloud (e.g. Hollenbach & Tielens, 1997, for a review). Because of their high density and their high $G_0$, they absorb a significant fraction of the FUV radiation by OB stars and reradiate it in the IR, as dust thermal emission and fine structure lines. Since they are the regions where molecules are being dissociated by FUV photons and subsequently recombine, they are the place of many important chemical reactions. Fig. III.51 shows the structure of a typical PDR. We have performed an isobaric run with the Meudon PDR code (Le Petit et al., 2006), for $P_{gas} = 10^7$ K/cm$^3$ and $G_0 = 10^5$. We show the variation of the abundances of the main species, and we represent in the upper part the H$_1$/H$_2$ and C$_{II}$/C$_I$/CO transitions. This figure demonstrates that H$_2$, being efficiently self-shielded, exists at lower A(V), whereas CO appears deeper into the cloud.

**Dense molecular clouds.** Dense molecular clouds ($n_{gas} \approx 10^3 - 10^6$ cm$^{-3}$; Table III.6) contain most of the molecular gas, concentrated in a small volume. These molecular clouds are gravitationally bound. Their structure is clumpy and filamentary. The gas motions are controlled by turbulence. They can be arranged in molecular complexes of sizes up to $\approx 100$ pc (e.g. Solomon et al., 1987). The densest cores are collapsing and will lead to star formation. One of the most challenging issue of ISMology is the difficulty to measure the mass of molecular clouds. As we have mentioned earlier, H$_2$ is a symmetric molecule. It thus does not have any dipole moment allowing rotational transitions. Its first transitions are its rovibrational levels (H$_2$ 0-0S(0)$_{28.3\mu m}$, H$_2$ 0-0S(1)$_{17.0\mu m}$ and so on) that need temperatures of a few hundred K to be pumped. Cold molecular clouds are thus primarily traced by the second most abundant molecule, CO, which is asymmetric. CO rotational lines, $^{12}$CO(J=1$\rightarrow$0)$_{2.6mm}$ and $^{12}$CO(J=2$\rightarrow$1)$_{1.3mm}$, are the most commonly observed transitions. The conversion of an observed $^{12}$CO(J=1$\rightarrow$0)$_{2.6mm}$ line intensity, $I_{CO}$, to a H$_2$ column density requires the knowledge of a CO-to-H$_2$ conversion factor, $X_{CO}$, such that (e.g. Bolatto et al., 2013):

$$N(H_2) \approx X_{CO} \times I_{CO}. \qquad (III.53)$$

The X(CO) factor has been calibrated on Galactic molecular clouds: $X(CO) \approx 2 \times 10^{20}$ cm$^{-2}$(K.km/s)$^{-1}$. This value is however metallicity dependent, as we will see in Sect. III.3.2.3.

### III.3.2 Dust as a Diagnostic Tool

We have already discussed the potential of dust as a tracer of the physical conditions in the ISM, especially in Sect. III.2.1.3. We review here a few examples where dust tracers were used to refine the results of studies dedicated to star formation or gas physics.

#### III.3.2.1 Dust to Study Star Formation

**Star formation rates.** Dust-related SFR tracers rely on the fact that young stars are extremely luminous and are enshrouded with dust. If the clouds are optically thick and if their covering factor is unity, the OB star luminosity is: $L_{OB} \approx L_{TIR}$. Contrary to a common misconception, this is independent of dust properties. Assuming a typical IMF, burst age and metallicity, $L_{OB}$ can be converted to: SFR $\approx 10^{-10} \times L_{TIR}/L_{\odot}$ (e.g. Kennicutt, 1998a). The contribution of old stars to $L_{TIR}$ is negligible for high enough SFRs. Alternatively, monochromatic SFR indicators have been proposed. Calzetti et al. (2007) and Li et al. (2010) found that the 24 and 70 µm monochromatic luminosities were good local SFR indicators (on spatial scales of $\approx 0.5 - 1$ kpc): SFR $\approx 2611 \times [L_{\odot}(24 \mu m)/(L_{\odot}/Hz)]^{0.885}$ and SFR $\approx 1547 \times L_{\odot}(70 \mu m)/(L_{\odot}/Hz)$. We have discussed the use of aromatic features as SFR tracers in Sect. III.2.1.3. Finally, several composite indicators have been calibrated (Hao et al., 2011). By combining FUV or H$\alpha_{656.3nm}$ measurements with the 24 µm or TIR indicators, they account for the fact that star-forming regions are not completely opaque.

<figure><img src="image_396.png" /></figure>
<figure><img src="image_397.png" /></figure>

<figure><img src="image_398.png" /></figure>

FIGURE III.51 – *Structure of a PDR.* The top drawing schematically represents the structure of a PDR. The ionizing stars are illuminating the cloud from the left. In the H II region, H is fully ionized. We enter the PDR at the ionization front. H is essentially neutral and atomic in this layer. We then pass the dissociation front around A(V) ≈ 0.5–1, where H is essentially molecular (H₂). Around A(V) ≈ 1.5–2, C becomes neutral, and around A(V) ≈ 3, CO becomes the dominant carbon-bearing species. We are at this stage in the cold molecular cloud. The two bottom panels show the results of an isobaric PDR model for P<sub>gas</sub> = 10<sup>7</sup> K/cm<sup>3</sup> and G<sub>0</sub> = 10<sup>5</sup> (using the Meudon PDR code; Le Petit et al., 2006). Panel (a) shows the evolution as a function of A(V) of the densities of the most important species. These densities are normalized by the total H density, n<sub>H</sub> = n(H<sup>+</sup>) + n(H<sup>0</sup>) + 2n(H<sub>2</sub>). Panel (b) shows the evolution of the gas and dust temperatures. The exact densities, A(V) and temperatures are specific to the particular model we have run, but they give a rough idea of the typical values of these parameters. Licensed under CC BY-SA 4.0.

<figure><img src="image_399.png" /></figure>
<figure><img src="image_400.png" /></figure>

**Resolving star formation.** Hony et al. (2015) performed a comparison of different SFR estimators with the actual stellar content of the star-forming region N66, in the SMC. In this study, we derived the stellar surface density, $\Sigma_\star$, based on individual star counts from HST photometry. When compared to the dust mass surface density, $\Sigma_{\text{dust}}$, derived from SED modeling, we found a significant scatter, at $\simeq 6$ pc linear resolution. The SFRs derived from $\Sigma_{\text{dust}}$ or $H\alpha_{656.3\text{nm}}$ underestimate the more reliable, $\Sigma_\star$-derived SFR, by up to a factor of $\simeq 10$. This is likely due to ionizing photons escaping the region$^{11}$. Finally, converting our $\Sigma_{\text{dust}}$ map to a gas mass surface density map, $\Sigma_{\text{gas}}$, we found that the pixels of our region are lying above the *Schmidt-Kennicutt relation*$^{12}$. This might be due to low density gas, inefficient at forming stars, that is included in global Schmidt-Kennicutt relations, but absent when zooming on star-forming regions.

### III.3.2.2 Photodissociation Regions

**PDR Properties.** We have participated to numerous studies aiming at constraining the PDR properties in low-metallicity environments (Cormier et al., 2010, 2012; Lebouteiller et al., 2012; Cormier et al., 2015; Chevance et al., 2016; Lebouteiller et al., 2017; Wu et al., 2018a; Cormier et al., 2019; Lebouteiller et al., 2019). The common point of these studies is their use of numerous fine structure lines observed by *Herschel* and *Spitzer*, as well as the dust emission, to constrain the main parameters of a PDR model ($G_0$, $n_{\text{gas}}$, filling factor). The challenge lies in the multiple degeneracies, due to the fact that a given line can trace several phases. For instance, $[\text{CII}]_{158\mu m}$ comes from the ionized gas, the neutral gas and from an important fraction of molecular clouds (Fig. III.51). Such a degeneracy can be solved by using additional lines to constrain the properties of these different phases. Dust tracers are also useful, either as a constraint or as a self-consistency test. As an example, Chevance et al. (2016) modeled the gas properties in 30 Doradus and derived the typical depth of PDRs, in terms of visual extinction magnitude, $A_{\text{PDR}}(V)$. Fig. III.52 shows the comparison of $A_{\text{PDR}}(V)$ to the value of this parameter, derived from SED modeling, $A_{\text{dust}}(V)$. We can see that both quantities are in good agreement, validating the PDR results.

**Photoelectric heating.** Assuming that $[\text{CII}]_{158\mu m}$ is the main gas coolant, the *photoelectric efficiency*, that we already discussed in Sect. III.3.1.1, $\epsilon_{\text{PE}}$, can be approximated by the gas-to-dust cooling ratio: $\epsilon_{\text{PE}} \simeq L_{\text{CII}}/L_{\text{TIR}}$. Detailed studies usually add other lines to the gas cooling rate, like $[\text{OI}]_{63\mu m}$, to have a more complete estimate (e.g. Lebouteiller et al., 2012; Cormier et al., 2015; Lebouteiller et al., 2019). Overall, Smith et al. (2017) found that $0.1\% \lesssim \epsilon_{\text{PE}} \lesssim 1\%$, with an average of $\langle\epsilon_{\text{PE}}\rangle \simeq 0.5\%$, in a sample of 54 nearby galaxies. It appears that $\epsilon_{\text{PE}}$ is lower when the dust temperature is higher (Rubin et al., 2009; Croxall et al., 2012). This is not likely the result of the destruction of the *UIB* carriers, as their intensity correlates the best with the $[\text{CII}]_{158\mu m}$ emission (e.g. Helou et al., 2001). It is rather conjectured to be due to the saturation of grain charging in *UV*-bright regions. The shape of the *ISRF* also has a consequence on the accuracy with which $L_{\text{TIR}}$ represents the true *UV*, photoelectric-efficient flux. Indeed, Kapala et al. (2017) showed that the variations of $\epsilon_{\text{PE}}$ in M 31 could be explained by the contribution of old stars to $L_{\text{TIR}}$. Finally, one of the most puzzling features is that $\epsilon_{\text{PE}}$ is higher at low metallicity (Poglitsch et al., 1995; Madden et al., 1997; Cormier et al., 2015; Smith et al., 2017; Madden et al., 2020), while the *UIB* strength drops in these systems (Sect. IV.2.2.1). This is currently poorly understood. However, in the extreme case of I Zw 18 ($Z \simeq 1/35 Z_\odot$), where no *UIB* is detected (e.g. Wu et al., 2006) and the photoelectric heating is estimated to be negligible, the gas-cooling-to-TIR ratio is still $\simeq 1\%$ (Lebouteiller et al., 2017). In this instance, we have shown the gas could be heated by X-rays.

<footnote n="11">11. We saw a similar discrepancy in the center of M 83, where the $[\text{NII}]_{122\mu m}$ line was significantly more extended than other SFR tracers (Wu et al., 2015).</footnote>

<footnote n="12">12. The Schmitt-Kennicutt relation is the empirical correlation between $\Sigma_{\text{SFR}}$ and $\Sigma_{\text{gas}}$ for a wide diversity of galaxies (Schmidt, 1959; Kennicutt, 1998b).</footnote>
<figure><img src="image_401.png" /></figure>

FIGURE III.52 – *Comparison of visual extinctions in 30 Doradus.* Panel (a) shows a map of 30 Doradus, seen through: (i) [SIV]10.51µm (red; heavily-ionized gas; Indebetouw et al., 2009); (ii) [NeII]12.81µm (green; moderately ionized gas; Indebetouw et al., 2009); (iii) [CII]158µm (blue; neutral gas; Chevance et al., 2016); and (iv) 12CO(J=3→2)867µm (white contours; molecular gas; Minami-dani et al., 2011). At the center of the image lies the SSC, R136. Panel (b) shows the comparison between the estimates of the visual extinction magnitudes, A(V) (Eq. II.4), in different regions of panel (a). The x-axis shows the A(V) value derived from the PDR modeling of the available gas lines, whereas the y-axis shows the value of A(V) inferred from dust SED modeling. Credit: (a) Chevance et al. (2016); (b) licensed under CC BY-SA 4.0.

### III.3.2.3 The Molecular Gas and its Dark Layer

**The dark gas.** We have discussed in Sect. III.3.1.3 that the photodissociation of H₂ and CO at different depths into molecular clouds leads to biases in gas mass estimates. H₂ is indeed self-shielded. It therefore exists at column densities roughly independent of metallicity. On the contrary, CO, which is significantly less abundant, is not self-shielded (i.e. its electronic lines remain optically thin at large column densities). Consequently, CO needs to be shielded by dust to survive. It thus exists only deeper into molecular clouds. The H₂ gas that exists in regions where CO is photodissociated is often referred to as the *CO-dark molecular gas*. Other tracers can be used to constrain this dark gas: (i) dust emission (e.g. Israel, 1997; Leroy et al., 2011); (ii) [CII]158µm (e.g. Madden et al., 1997); (iii) and γ-rays (e.g. Grenier et al., 2005). Using I<sub>CO</sub> to derive N(H<sub>2</sub>) with Eq. (III.53) can therefore be biased if the dark gas fraction is significantly larger than in the MW, where X(CO) has been calibrated. This is what happens in low-metallicity systems, where the *dustiness* is lower, because of the lower abundance of heavy elements (cf. Sect. IV.3.1). This is represented in Fig. III.53. We see that in the low-metallicity cloud (on the right), CO cores are much smaller, because of the increased photodissociation of this molecule, compared to the left cloud (Solar metallicity). It results that the mass of CO-dark H₂ is significantly larger at low metallicity. We have investigated this ISM component in the LMC, using dust mass surface density, concluding it could account between ≈ 10% and ≈ 100% of the total molecular gas mass (Galliano et al., 2011). Recently, we studied dark gas in the star-forming region N11 of the LMC, modeling the full set of IR emission lines (Lebouteiller et al., 2019). We showed that most of
the molecular gas in this region is CO-dark and that [CII]158µm traces mostly this component. We extended this analysis to a sample of nearby dwarf galaxies (Madden et al., 2020). We found that ≈ 70–100% of the molecular gas mass is not traced by $^{12}$CO(J=1→0)2.6mm.

<figure><img src="image_402.png" /></figure>

The molecular gas content of low-metallicity systems is dominated by dark gas.

Solar Metallicity

<figure><img src="image_403.png" /></figure>

Young Stars

Low Metallicity

<figure><img src="image_404.png" /></figure>

FIGURE III.53 – *Metallicity effect on the CO-dark gas.* We have represented two molecular clouds photodissociated by nearby OB star associations. The left cloud has a Solar metallicity. Its CO core is efficiently shielded by dust. This is not the case for the right cloud, which has a low metallicity. In this cloud, the lower grain abundance causes a lower dust screening. Consequently, CO is photodissociated deeper into the cloud, whereas H$_2$ remains self-shielded. Credit: Madden et al. (2020).

**Pressure and radiation field.** The pressure in molecular clouds can be significantly larger than in the pressure equilibrium phases of the ISM: $P_{\text{gas}} \approx 6 \times 10^7 \text{ K/cm}^{-3}$ in the Orion bar (Goicoechea et al., 2016); compared to $P_{\text{gas}} \approx 3000 \text{ K/cm}^{-3}$ in the HIM, WIM and CNM (Eq. III.46). We have studied the physical conditions of the molecular gas in the central region of the starbursting galaxy, M 83 (Wu et al., 2015). We used the CO Spectral Line Energy Distribution (SLED) observed by Herschel to estimate its column density and pressure, N(CO) and $P_{\text{CO}}$, in different regions. We have also performed SED modeling to estimate the mean starlight intensity heating the grains, $\langle U \rangle$ (Eq. III.38). This allowed us to show that both quantities are correlated (cf. Fig. III.54.a). We also noted that the pressure gradient was oriented along a chain of radio sources, corresponding to a radio jet (Fig. III.54.b). We derived a similar correlation between the ISRF strength and the gas thermal pressure in the Carina nebula (Wu et al., 2018a). Such a correlation was also found by Joblin et al. (2018) in the Orion bar. They argue that the photoevaporation of the PDR can explain this relation.
<figure><img src="image_405.png" /></figure>

<figure><img src="image_406.png" /></figure>

<figure><img src="image_407.png" /></figure>

FIGURE III.54 – *Molecular gas pressure in the center of M 83.* Panel (a) displays the relation between the average starlight intensity derived from SED modeling, and the molecular gas pressure, inferred from CO SLED modeling, in different regions in the center of M 83 (Wu et al., 2015). Panel (b) shows the SFR map of the central region of M 83. The color contours represent the molecular gas pressure, from $P_{CO} = 4 \times 10^{5}$ K/cm$^{3}$ (blue) to $P_{CO} = 2.4 \times 10^{6}$ K/cm$^{3}$ (red). The four diamonds indicates the position of the four radio sources reported by Maddox et al. (2006). Credit: (a) licensed under CC BY-SA 4.0; (b) Wu et al. (2015).

<figure><img src="image_408.png" /></figure>
<figure><img src="image_409.png" /></figure>

# Chapter IV

# Modeling Cosmic Dust Evolution

This is not a new result – Draine & Salpeter (1979) reached the conclusion that grain destruction was rapid and that regrowth of dust in the ISM was required to explain the observed depletions. The numbers basically haven't changed appreciably since then; the argument has been reiterated a number of times (...). Nevertheless, some authors continued to hold the view that the solids in the interstellar medium were primarily formed in stars.

(Bruce T. DRAINE; Draine, 2009)

## Contents

<table><tr><td>IV.1 Stellar Evolution</td><td>180</td></tr><tr><td>IV.1.1 The Fate of Stars of Different Masses</td><td>180</td></tr><tr><td>IV.1.1.1 Nucleosynthesis</td><td>180</td></tr><tr><td>Nuclear binding energies.</td><td>180</td></tr><tr><td>Primordial nucleosynthesis.</td><td>180</td></tr><tr><td>Stellar nucleosynthesis.</td><td>180</td></tr><tr><td>IV.1.1.2 Brief Outline of Stellar Evolution</td><td>181</td></tr><tr><td>The Main Sequence.</td><td>181</td></tr><tr><td>The late stages of massive stars.</td><td>182</td></tr><tr><td>The late stages of LIMS.</td><td>182</td></tr><tr><td>IV.1.1.3 Parametrizing Star Formation</td><td>183</td></tr><tr><td>Initial mass functions.</td><td>183</td></tr><tr><td>Parametric star formation histories.</td><td>183</td></tr><tr><td>IV.1.2 Elemental and Dust Yields</td><td>185</td></tr><tr><td>IV.1.2.1 Injection of Heavy Elements in the ISM</td><td>186</td></tr><tr><td>Stellar elemental yields.</td><td>186</td></tr><tr><td>Metallicity estimates.</td><td>188</td></tr><tr><td>IV.1.2.2 Production of Stardust</td><td>188</td></tr><tr><td>AGB stars.</td><td>188</td></tr><tr><td>SN II.</td><td>188</td></tr><tr><td>Indirect evidence.</td><td>189</td></tr><tr><td>IV.2 Dust Evolution Processes in the ISM</td><td>189</td></tr><tr><td>IV.2.1 Grain Formation and Transformation</td><td>189</td></tr><tr><td>IV.2.1.1 Evidence of Grain Growth and Coagulation in the ISM</td><td>190</td></tr><tr><td>The evidence brought by depletions.</td><td>190</td></tr><tr><td>FIR opacity variations.</td><td>190</td></tr></table>

<figure><img src="image_410.png" /></figure>
<table>
  <tbody>
    <tr>
      <td>IV.2.1.2</td>
      <td>Studies of the Magellanic Clouds</td>
      <td>190</td>
    </tr>
    <tr>
      <td></td>
      <td>Spatially-resolved SED fitting of LMC-N 44 and SMC-N 66.</td>
      <td>191</td>
    </tr>
    <tr>
      <td></td>
      <td>Derived dust-gas relations.</td>
      <td>191</td>
    </tr>
    <tr>
      <td>IV.2.1.3</td>
      <td>Quantifying Grain Growth</td>
      <td>192</td>
    </tr>
    <tr>
      <td></td>
      <td>Accretion timescale.</td>
      <td>193</td>
    </tr>
    <tr>
      <td></td>
      <td>Grain growth in different ISM phases.</td>
      <td>194</td>
    </tr>
    <tr>
      <td></td>
      <td>Relation to global parameters.</td>
      <td>194</td>
    </tr>
    <tr>
      <td>IV.2.2</td>
      <td>Grain Destruction.</td>
      <td>195</td>
    </tr>
    <tr>
      <td>IV.2.2.1</td>
      <td>Photodestruction of Small Grains</td>
      <td>195</td>
    </tr>
    <tr>
      <td></td>
      <td>Photodesorption and sublimation.</td>
      <td>195</td>
    </tr>
    <tr>
      <td></td>
      <td>Evidence in resolved regions.</td>
      <td>196</td>
    </tr>
    <tr>
      <td>IV.2.2.2</td>
      <td>Thermal Sputtering</td>
      <td>196</td>
    </tr>
    <tr>
      <td></td>
      <td>Sputtering times.</td>
      <td>198</td>
    </tr>
    <tr>
      <td></td>
      <td>Early-type galaxies.</td>
      <td>199</td>
    </tr>
    <tr>
      <td>IV.2.2.3</td>
      <td>Destruction by SN Blast Waves</td>
      <td>199</td>
    </tr>
    <tr>
      <td></td>
      <td>Evidence in resolved regions.</td>
      <td>200</td>
    </tr>
    <tr>
      <td></td>
      <td>Global model prescription.</td>
      <td>200</td>
    </tr>
    <tr>
      <td></td>
      <td>The destruction efficiency.</td>
      <td>200</td>
    </tr>
    <tr>
      <td>IV.3</td>
      <td>Cosmic Dust Evolution</td>
      <td>201</td>
    </tr>
    <tr>
      <td>IV.3.1</td>
      <td>Constraining the Dust Build-Up in Galaxies</td>
      <td>201</td>
    </tr>
    <tr>
      <td>IV.3.1.1</td>
      <td>Cosmic Dust Evolution Models.</td>
      <td>202</td>
    </tr>
    <tr>
      <td></td>
      <td>Physical ingredients and assumptions.</td>
      <td>202</td>
    </tr>
    <tr>
      <td></td>
      <td>The ejected masses.</td>
      <td>203</td>
    </tr>
    <tr>
      <td></td>
      <td>The equations of evolution.</td>
      <td>203</td>
    </tr>
    <tr>
      <td></td>
      <td>Dust evolution tracks.</td>
      <td>204</td>
    </tr>
    <tr>
      <td></td>
      <td>Effects of the individual parameters.</td>
      <td>204</td>
    </tr>
    <tr>
      <td>IV.3.1.2</td>
      <td>Empirical Inference of Dust Evolution Timescales</td>
      <td>206</td>
    </tr>
    <tr>
      <td></td>
      <td>Fitting dust scaling relations.</td>
      <td>206</td>
    </tr>
    <tr>
      <td></td>
      <td>Evolutionary timescales as a function of metallicity.</td>
      <td>208</td>
    </tr>
    <tr>
      <td></td>
      <td>Methodological remarks.</td>
      <td>210</td>
    </tr>
    <tr>
      <td></td>
      <td>The controversy about stardust.</td>
      <td>211</td>
    </tr>
    <tr>
      <td></td>
      <td>Limitations of our approach.</td>
      <td>212</td>
    </tr>
    <tr>
      <td>IV.3.2</td>
      <td>Evolution of the Aromatic Feature Carriers</td>
      <td>212</td>
    </tr>
    <tr>
      <td>IV.3.2.1</td>
      <td>The Different Evolution Scenarios</td>
      <td>213</td>
    </tr>
    <tr>
      <td></td>
      <td>Enhanced destruction at low metallicity.</td>
      <td>213</td>
    </tr>
    <tr>
      <td></td>
      <td>Inhibited formation efficiency at low metallicity.</td>
      <td>213</td>
    </tr>
    <tr>
      <td>IV.3.2.2</td>
      <td>The Observed Trends</td>
      <td>214</td>
    </tr>
    <tr>
      <td></td>
      <td>A better correlation with metallicity.</td>
      <td>214</td>
    </tr>
    <tr>
      <td></td>
      <td>The global point of view.</td>
      <td>214</td>
    </tr>
    <tr>
      <td></td>
      <td>The potential of quiescent very-low-metallicity galaxies.</td>
      <td>215</td>
    </tr>
  </tbody>
</table>

This chapter focusses on the study of dust evolution in all interstellar environments, at all spatial scales. Dust evolution is the variation of the constitution of a grain mixture with time, under the effects of its environment. The timescales of evolution being significantly longer than the career of a scientist, we usually study spatial variations of the dust content in a region, or the variations among a sample of galaxies. These different observations are then compared, being considered as snapshots at different evolutionary stages. The environmental parameters that are commonly used to quantify dust evolution are: (i) the ISM density and the ISRF intensity and hardness, for spatially-resolved studies; (ii) the metallicity and star formation rate, for global galactic studies. The main processes responsible for dust evolution are represented on Fig. IV.1.
<figure><img src="image_411.png" /></figure>

**Grain Formation** is the dust mass build-up by:

* grain condensation in the ejecta of core-collapse SNe and AGB stars (i.e. making grains from scratch by condensing atoms);

* grain (re-)formation in the CNM and molecular clouds, by accretion of atoms and molecules onto grains: (i) grain growth; (ii) mantle accretion; and (iii) ice formation.

**Grain Processing** is the alteration of the grain constitution in the ISM by:

* shattering and fragmentation by grain-grain collisions in low-velocity shocks (modification of the size distribution);

* structural modifications by high energy photons or cosmic ray impacts;

* grain-grain coagulation in cold regions.

**Grain Destruction** is the full or partial removal of the elements constituting the grains by:

* erosion and evaporation by thermal or kinetic sputtering (gas-grain collision in a hot gas or a shock);

* photodesorption of atoms and molecules;

* thermal evaporation;

* astration (incorporation into stars).

<figure><img src="image_412.png" /></figure>

FIGURE IV.1 – *The interstellar dust lifecycle.* This is the schematic representation of dust evolution through the ISM. The upper part illustrates the different sources, sinks and environments where grains are processed. The timescales are indicative and will be discussed in the rest of this chapter. The eight small bottom panels focus on the microscopic processes. Licensed under CC BY-SA 4.0.
<figure><img src="image_413.png" /></figure>

# IV.1 Stellar Evolution

Stars have a crucial impact on ISD: (i) they synthesize the heavy elements that constitute dust grains (Fig. II.16); (ii) they also directly produce dust seeds in their ejecta; (iii) the shock waves of SNe erode and vaporize the grains; (iv) the radiative and mechanical feedback of massive stars carve the ISM and process the grains.

## IV.1.1 The Fate of Stars of Different Masses

A star can be conceptualized as a sphere of gas in hydrostatic equilibrium, where the gravity is counterbalanced by the thermal pressure sustained by nuclear reactions in its core (e.g. Degl'Innocenti, 2016, for an introduction). The energy produced in the core is carried out through radiation, convection or conduction. The initial mass of a star, and in a lesser extent its initial metallicity, determine its future evolution.

### IV.1.1.1 Nucleosynthesis

The nuclear reactions in stellar interiors, on top of being the fuel of stars, lead to the production of heavy elements. A fraction of these freshly synthesized elements are injected back into the ISM, during the final stages of stellar evolution.

**Nuclear binding energies.** A fundamental quantity to determine the efficiency of nuclear reactions to sustain the thermal pressure within a star is the nuclear binding energy of an element of mass A (number of nucleons; cf. e.g. Chaps. 1-2 of Pagel, 1997, for a review). This quantity is represented in Fig. IV.2 for the most relevant nuclei. To have an exothermic reaction, that will be able to counterbalance gravity, one needs to synthesize elements of higher binding energies. The curve of Fig. IV.2 reaches a maximum around $^{56}$Fe.

**Fusion** of elements lighter than $^{56}$Fe is exothermic. Since the initial composition of a star is $\approx 3/4$ H and $\approx 1/4$ He, stellar nucleosynthesis takes this way.

**Fission** of elements heavier than $^{56}$Fe is exothermic. This is the process implemented in nuclear reactors to generate electricity (through heat).

**Primordial nucleosynthesis.** Before the first stars appeared, $^{2}$H and $^{4}$He, as well as elements up to $^{7}$Li, were synthesized during the first 15 minutes after the big bang (e.g. Pagel, 1997; Calura & Matteucci, 2004; Johnson, 2019). The temperature was at this time around $T \approx 10^9$ K. This primordial nucleosynthesis was brief, as the Universe was expanding and cooling. It is estimated that after 20 minutes, the temperature was too low to synthesize new elements. The *primordial abundances* refer to the elements produced during these first minutes (cf. Eq. II.11):

$$X_{\text{primordial}} \approx 0.76, \quad Y_{\text{primordial}} \approx 0.24, \quad Z_{\text{primordial}} \approx 0.00. \qquad (\text{IV.1})$$

**Stellar nucleosynthesis.** Once the temperature at the center of a collapsing protostar becomes high enough ($T \approx 10^7$ K), thermonuclear reactions<fnref n="1" /> are initiated. Several chains and cycles of reactions occur in stars, at different stages. The most important ones are the following (e.g. Filippone, 1986; Pagel, 1997; Silva Aguirre, 2018).

**Proton fusion**, also called *p-p chain*, is a series of nuclear reactions converting $^{4}$H into $^{4}$He. This reaction chain has three branches cycling through various light elements (D, Li, B, Be). It is the dominant process in stellar interiors with $T \lesssim 2 \times 10^7$ K, that is for stars with mass $m_{\star} \lesssim 1$ M$_{\odot}$.<fnref n="2" />

<figure><img src="image_414.png" /></figure>

<footnote n="1">1. In thermonuclear reactions, the high temperature gives nuclei enough kinetic energy to overcome their Coulomb barrier, and allows them to fuse with each other.</footnote>

<footnote n="2">2. The Sun's core is at $T \approx 1.5 \times 10^7$ K.</footnote>
<figure><img src="image_415.png" /><figcaption>FIGURE IV.2 – *Average nuclear binding energies per nucleon.* We have displayed the experimental data compiled by Ghahramany et al. (2012). The most stable elements are around $^{56}$Fe. Below, fusion is exothermic, above, it is fission. Licensed under CC BY-SA 4.0.</figcaption></figure>

**CNO cycle** is another series of nuclear reactions converting $4^1\text{H}$ into $4^1\text{He}$. Contrary to the p-p chain, this cycle requires pre-existing C, N or O (i.e. it requires a non-zero-metallicity star). This cycle can be broken into: (i) a *CN cycle*, starting with $^{12}\text{C}$ and $^{4}\text{H}$, ending with $^{12}\text{C}$ and $^{4}\text{He}$; and (ii) a *NO cycle*, starting with $^{15}\text{N}$ and $^{3}\text{H}$, ending with $^{14}\text{N}$ and $^{4}\text{He}$. This cycle is more efficient than the p-p chain for $T \gtrsim 2 \times 10^7$ K, that is for stars with $m_{\star} \gtrsim 1$ M$_\odot$. In practice, both happen simultaneously, but with different efficiencies.

**The triple α process** is a series of nuclear reactions converting $3^4\text{He}$ into $^{12}\text{C}$ (the $^4\text{He}$ nucleus is indeed called the *α particle*). It also produces $^{16}\text{O}$ and $^{20}\text{Ne}$ as byproducts. This process starts when the star has converted ≈ 10% of its H into He. It requires temperatures of $T \approx 10^8$ K.

**Heavier element fusion** happens essentially in massive stars ($m_{\star} \gtrsim 8$ M$_\odot$), when the temperature of the core reaches $T \approx 10^9$ K. Several successive phases are then possible: C burning, Ne burning and O burning, producing up to $^{28}\text{Si}$. The last series of reactions are the α ladder, which produce elements up to Fe and Ni.

☞ *H burning*, which encompasses both the p-p chain and the CNO cycle, represents the longest phase in the lifetime of a star, whereas *He burning* lasts only ≈ 10% of its existence.

### IV.1.1.2 Brief Outline of Stellar Evolution

Stars are born from the collapse of molecular clouds into protostars (e.g. Motte et al., 2018, for a review). Protostars accrete matter until their winds and radiation pressure stops this process, leading to a *pre-main sequence star* (pre-MS). Pre-MS stars exhibit violent winds and bipolar jets, clearing away the remaining molecular cocoon they were born in. They contract until the temperature in their core is high enough to initiate H fusion ($T \approx 10^7$ K). Below $m_{\star} \le 0.08$ M$_\odot$, we get a *brown dwarf*, which is a compact object not massive enough to sustain nuclear reactions. Fig. IV.3 schematically represents the different stages of evolution of low- and high-mass stars.

**The Main Sequence.** Once nuclear reactions ignite, stars are on the *Zero-Age Main Sequence* (ZAMS; grey line in Fig. III.18). We have already briefly discussed stellar evolution in Sect. III.1.2.3. The dif-
<figure><img src="image_416.png" /></figure>

ferent types of stars, their mass, luminosities and lifetimes are given in Fig. III.18 and in Table III.2.

1. As long as stars are in their H burning phase (cf. Sect. IV.1.1.1) they are *Main Sequence* (MS) stars. They move slowly along their tracks in Fig. III.18.

2. Once the core has exhausted its H, it contracts by lack of fuel. This contraction increases the temperature, allowing the He burning to start (cf. Sect. IV.1.1.1). Due to the increase of central temperature, the outer layers expand. The star is now a *red giant*.

3. This process repeats, with He burning (triple α process; Sect. IV.1.1.1).

**The late stages of massive stars.** Massive stars ($8 M_{\odot} \le M_{\star} < 120 M_{\odot}$) are the hottest and most luminous ones (cf. Table III.2). They are short-lived ($\tau(m_{\star}) \lesssim 30$ Myr; Fig. III.18).

1. The process of burning heavier elements is repeated beyond C, resulting in an “onion” structure (cf. Fig. IV.3.a). The combustion of each element is exponentially faster. These stars are, at this point, *red supergiants* (cf. Fig. III.18).

2. Once the core is made of Fe, the star can not anymore produce energy by nuclear fusion. It therefore collapses (e.g. Heger et al., 2003).

If $m_{\star} \lesssim 40 M_{\odot}$, the collapse is halted by the *degeneracy pressure*[^3] of neutrons. The outer layers of the stars then explode as a *type II supernova* (SN II) or *core-collapse supernova*, leaving a *Neutron Star* (NS) in the center.

If $m_{\star} \gtrsim 40 M_{\odot}$, the degeneracy pressure of the neutron core is not sufficient to sustain the collapse. The remnant is not anymore a NS, but a *Black Hole* (BH). This is also the approximate mass range where the star leaves a remnant without exploding as a SN, ending as a *collapsar* (Heger et al., 2003). These two phenomena (ending as a collapsar and leaving a BH) are not necessarily concomitant. The exact masses above which these two phenomena occur are not accurately known and depend on other parameters, such as stellar rotation. For simplicity, we have represented both phenomena on the same branch in Fig. IV.3.

**The late stages of LIMS.** *Low- and Intermediate-Mass Stars* (LIMS; $0.08 M_{\odot} \le M_{\star} < 8 M_{\odot}$) are less luminous than massive stars, but they are the most numerous. Their lower gravity allow them to burn their elements slower than massive stars, and therefore to live longer (several Gyrs, on average; Fig. III.18).

1. Their mass does not allow them to start C burning. LIMS enter the *Asymptotic Giant Branch* (AGB; cf. Fig. III.18; e.g. van Winckel, 2003; Herwig, 2003, for reviews). They are larger and more luminous than red giants and are thermally pulsing.

2. The contraction of the core is halted by electron degeneracy pressure. The thermal pulses lead the outer shell to expand progressively, creating a *Planetary Nebula* (PN), leaving a *White Dwarf* (WD) in the center.

3. The maximum mass a white dwarf can reach is the *Chandrasekar mass*, $m_{\text{Chandra}} \approx 1.4 M_{\odot}$. If a white dwarf of mass $m_{\text{WD}}$ happens to be in a binary system with another red giant, it will accrete some of its mass.

If $m_{\text{WD}} \gtrsim m_{\text{Chandra}}$, the excess of mass above the Chandrasekar limit re-ignites the thermonuclear reactions. It follows a *type Ia SN* (SN Ia), which disrupts the binary system.

If $m_{\text{WD}} \lesssim m_{\text{Chandra}}$, thermonuclear reactions are ignited at the surface of the white dwarf. It ensues a *nova*, that does not disrupt the binary system.

<figure><img src="image_417.png" /></figure>

<footnote n="3">3. The degeneracy pressure is due to the fact that fermions can not occupy the same state. In very dense environments, this leads to a pressure: electron degeneracy pressure in white dwarfs; neutron degeneracy pressure in neutron stars.</footnote>
<figure><img src="image_418.png" /></figure>

#### IV.1.1.3 Parametrizing Star Formation

Star formation is a complex process involving stars of different masses being formed at different times. At the scale of a star-forming region or an entire galaxy, SF can be described statistically.

**Initial mass functions.** Initial Mass Functions (IMF) express the number distribution of stars of mass $m_\star$ born at a given time: $\phi(m_\star) \equiv dN_\star/dm_\star$. IMFs are usually expressed in $M_\odot^{-1}$, and are normalized as $^4$:

$$ \int_{m_{-}}^{m_{+}} \phi(m_{\star}) dm_{\star} = 1, \qquad (IV.2) $$

where $m_{-} = 0.1 M_\odot$ and $m_{+} = 100 M_\odot$ are the lower and upper masses. The average stellar mass is defined as:

$$ \langle m_{\star} \rangle \equiv \int_{m_{-}}^{m_{+}} m_{\star} \phi(m_{\star}) dm_{\star}. \qquad (IV.3) $$

The fraction of stars ending their life as a core-collapse SN is:

$$ f_{\text{SN}} \equiv \int_{m_{-}}^{m_{+}} m_{\star}^{\text{SN}} \phi(m_{\star}) dm_{\star}, \qquad (IV.4) $$

with $m_{-}^{\text{SN}} = 8 M_\odot$ and $m_{+}^{\text{SN}} = 40 M_\odot$ (e.g. Heger et al., 2003). Several IMFs have been proposed in the literature (see also Kroupa, 2001).

**The Salpeter IMF** (Salpeter, 1955) was the first one proposed. It is defined as:

$$ \phi_{\text{Salp}}(m_{\star}) \equiv \frac{(1-\alpha) \times m_{\star}^{-\alpha}}{m_{+}^{1-\alpha} - m_{-}^{1-\alpha}}, \quad \text{with } \alpha = 2.35 \qquad (IV.5) $$

where the lower and upper masses, $m_{-}$ and $m_{+}$, are usually taken as $m_{-} = 0.1 M_\odot$ and $m_{+} = 100 M_\odot$, although the original Salpeter (1955) study constrained the index of the power-law only up to $m_{+} = 10 M_\odot$.

**The Chabrier IMF** (Chabrier, 2003) for individual stars is defined, within the same mass range, as:

$$ \phi_{\text{Chab}}(m_{\star}) \equiv \begin{cases} 0.16033 \times m_{\star}^{-2.3} & \text{for } m_{\star} > 1 M_{\odot} \\ \frac{0.5718}{m_{\star}} \exp\left(-\frac{(\log_{10}(m_{\star}) + 1.1023729)^2}{0.9522}\right) & \text{for } m_{\star} \le 1 M_{\odot}. \end{cases} \qquad (IV.6) $$

**The top-heavy IMF** (e.g. Dwek et al., 2007) is often invoked at high-redshift. It is defined as:

$$ \phi_{\text{Top}}(m_{\star}) \equiv \frac{(1-\alpha) \times m_{\star}^{-\alpha}}{m_{+}^{1-\alpha} - m_{-}^{1-\alpha}}, \quad \text{with } \alpha = 1.50 \qquad (IV.7) $$

These IMFs are compared in Fig. IV.4. Some of their properties are listed in Table IV.1. The IMF is thought to be a universal property of interstellar media. The different IMFs of Table IV.1 have consequences on the stellar properties. The current consensus is that the Chabrier IMF might be more appropriate than Salpeter's, at least at low redshift.

**Parametric star formation histories.** We have already briefly discussed Star Formation Histories (SFH) in Sect. III.1.2.3. The SFH quantifies the SFR, $\psi(t)$, as a function of time, $t$, of a star-forming region or galaxy. Several parametric forms are commonly used in the literature. As we will see in Sect. IV.3, their parameters can be inferred by fitting a set of observations.

**The exponential SFH** (cf. Fig. IV.5.a) is parametrized by a timescale, $\tau_{\text{SF}}$, and a SFR at $t=0$, $\psi_0$:

$$ \psi_{\text{exp}}(t) \equiv \psi_0 \exp\left(-\frac{t}{\tau_{\text{SF}}}\right). \qquad (IV.8) $$

<footnote n="4">4. Careful though, some authors quote IMFs, normalized as $\int m_\star \phi(m_\star) dm_\star = 1$.</footnote>
<figure><img src="image_419.png" /><figcaption>FIGURE IV.3 – Schematic representation of stellar evolution. Licensed under CC BY-SA 4.0.</figcaption></figure>

<table><caption>TABLE IV.1 – IMF properties. These properties are integrated over three different IMFs: Salpeter (Eq. IV.5); Chabrier (Eq. IV.6); and Top-heavy (Eq. IV.7). For the last three lines, we used the ZAMS stellar luminosities of Fig. III.18 (initial metallicity Z = 0.008).</caption><thead><tr><th></th><th>Salpeter</th><th>Chabrier</th><th>Top-Heavy</th></tr></thead><tbody><tr><td>Average mass, &lt;m<sub>★</sub>&gt;</td><td>0.351 M<sub>☉</sub></td><td>0.673 M<sub>☉</sub></td><td>3.16 M<sub>☉</sub></td></tr><tr><td>SN II fraction, f<sub>SN</sub></td><td>0.239%</td><td>0.724%</td><td>6.38%</td></tr><tr><td>Mass fraction of massive stars</td><td>13.9%</td><td>22.6%</td><td>74.1%</td></tr><tr><td>LIMS luminosity (t<sub>★</sub> = 0, Z = 0.008)</td><td>4.81 L<sub>☉</sub>/M<sub>☉</sub></td><td>13.9 L<sub>☉</sub>/M<sub>☉</sub></td><td>55.3 L<sub>☉</sub>/M<sub>☉</sub></td></tr><tr><td>Massive star luminosity (t<sub>★</sub> = 0, Z = 0.008)</td><td>180 L<sub>☉</sub>/M<sub>☉</sub></td><td>579 L<sub>☉</sub>/M<sub>☉</sub></td><td>1.38 &times; 10<sup>4</sup> L<sub>☉</sub>/M<sub>☉</sub></td></tr><tr><td>Total luminosity (t<sub>★</sub> = 0, Z = 0.008)</td><td>185 L<sub>☉</sub>/M<sub>☉</sub></td><td>593 L<sub>☉</sub>/M<sub>☉</sub></td><td>1.38 &times; 10<sup>4</sup> L<sub>☉</sub>/M<sub>☉</sub></td></tr></tbody></table>
<figure><img src="image_420.png" /></figure>

<figure><img src="image_421.png" /></figure>

FIGURE IV.4 – Initial mass functions. We compare the three IMFs discussed in the text: (i) Salpeter (1955) (Eq. IV.5); (ii) Chabrier (2003) (Eq. IV.6); and (iii) Top-heavy (Eq. IV.7). Panel (a) shows the number distribution of stars, $m_{\star}\phi(m_{\star})$. Panel (b) shows the luminosity distribution, $L_{\star}m_{\star}\phi(m_{\star})$, where $L_{\star}$ comes from Fig. III.18. We have indicated in yellow the range $8 \le m_{\star} < 40 M_{\odot}$, corresponding to SN II. Licensed under CC BY-SA 4.0.

**The delayed SFH** (cf. Fig. IV.5.b; Lee et al., 2010) has the same number of parameters as the exponential SFH, but its SFR peak is delayed at $t = \tau_{SF}$:

$$ \psi_{\text{del}}(t) = \psi_0 \frac{t}{\tau_{\text{SF}}} \exp\left(-\frac{t}{\tau_{\text{SF}}}\right). \qquad (\text{IV.9}) $$

It is possible to combine several SFHs to account for the complex history of a galaxy. A useful quantity, deriving from the SFH, is the stellar birth rate, which is the average number of stars born per unit time:

$$ B(t) = \frac{\psi(t)}{\langle m_{\star} \rangle}. \qquad (\text{IV.10}) $$

The rate of SN II, $R_{\text{SN}}(t)$, can be approximated from this quantity:

$$ R_{\text{SN}}(t) = \int_{m_{\star}^{\text{SN}}}^{m_{\star}^{\text{SN}}} B(t - \tau(m_{\star})) \times \phi(m_{\star}) dm_{\star} \approx B(t) f_{\text{SN}}, \qquad (\text{IV.11}) $$

where $\tau(m_{\star})$ is the lifetime of a star of mass $m_{\star}$ (cf. Fig. III.18). The approximation, in the second part of Eq. (IV.11), comes from the fact that the lifetime of massive stars ($\tau(m_{\star}) \lesssim 30$ Myr) is usually much smaller than the SF timescale: $\tau(m_{\star}) \ll \tau_{SF}$.

IV.1.2 Elemental and Dust Yields

Stars, in their late stages, return to the ISM a fraction of the heavy elements they have synthesized. Stellar ejecta are: (i) stellar winds; (ii) planetary nebulae; (iii) novae; (iv) SN Ia; and (v) SN II. In addition, the temperature in these ejecta can be low enough (cf. Fig. II.17.b) to condense grains, that we refer to as *stardust*.
<figure><img src="image_422.png" /></figure>

FIGURE IV.5 – *Parametric star formation histories.* Panel (a) represents the exponential SFH (Eq. IV.8), for different values of τ<sub>SF</sub>. Panel (b) represents the delayed SFH (Eq. IV.9), for the same grid of τ<sub>SF</sub>. In both panels, the SFHs are normalized so that the integrated mass of stars formed in Δt = 14 Gyr (about the age of the Universe) is M<sub>⋆</sub> = 4 × 10<sup>10</sup> M<sub>☉</sub> (roughly the MW stellar mass). Licensed under CC BY-SA 4.0.

IV.1.2.1 Injection of Heavy Elements in the ISM

Fig. IV.6 gives the approximate fraction of each element produced in different environments, for the Solar neighborhood (e.g. Johnson, 2019). These proportions depend on the past SFH of the system we are considering.

**Primordial nucleosynthesis** (cf. Sect. IV.1.1.1) is responsible for the production of most of the light elements.

SN II account for most of the ISD-relevant elements, except C.

**AGB ejecta** (i.e. LIMS winds and PNe) account for a significant fraction of C and N, and heavier elements we have not displayed here.

SN1a are responsible for synthesizing a significant fraction of the metals around Fe.

**Other processes**, such as cosmic ray fission and merging neutron stars, are not very relevant to ISD.

**Stellar elemental yields.** A stellar yield, Y<sub>E</sub>(m<sub>⋆</sub>), is the mass of an element E injected into the ISM by a star of mass m<sub>⋆</sub>, at the end of its lifetime. These yields can be constrained observationally, but they are essentially determined theoretically (e.g. Karakas & Lattanzio, 2014, for a review). Fig. IV.7.a shows the yields of the most important elements. An important quantity determining the type of dust grains that will form in the ejecta is the C/O ratio. Indeed, when the temperature cools down enough, C and O tend to combine to form CO molecules. The excess atom will thus be the only one left to form stardust. Therefore, stellar ejecta with C < O will form primarily O-rich grains (silicates and oxides), whereas stellar ejecta with C > O will form mainly carbon grains and SiC. Fig. IV.7.b compares the number abundances of C and O ejected by stars of different masses. We can see that carbon grains originate mainly in LIMS around m<sub>⋆</sub> ≈ 3 M<sub>☉</sub>.

☞ SN II are responsible for most O-rich stardust, while LIMS produce most C-rich grain seeds.
<figure><img src="image_423.png" /></figure>

FIGURE IV.6 – Nucleosynthesis origin of the main elements. This curve represents the Solar abundances of Asplund et al. (2009) that we have shown in Fig. II.16. We have indicated the fraction of the different sources of nucleosynthesis where these elements originate from (Table S1 of Johnson, 2019). Since we have stopped at Z = 36, none of the elements shown are coming from merging neutron stars. Licensed under CC BY-SA 4.0.

<figure><img src="image_424.png" /></figure>

FIGURE IV.7 – Solar metallicity elemental stellar yields. Panel (a) shows the mass of the most abundant elements ejected by stars of different masses. Panel (b) focusses on the stellar mass range where the number abundance of C exceeds that of O (in yellow). These data are the yields used by the chemical evolution model of Galliano et al. (2008a). They are an homogenization of the yields of Karakas & Lattanzio (2003a,b), for LIMS, and of Woosley & Weaver (1995), for high-mass stars. Licensed under CC BY-SA 4.0.
**Metallicity estimates.** The heavy elements injected by the successive stellar populations increase the metallicity of the ISM, Z. In that sense, the metallicity is an indicator of “the fraction of baryonic matter that has been converted into heavier elements by means of stellar nucleosynthesis” (Kunth & Östlin, 2000), such that:

$$ \left\{ \begin{array}{l} X_{\text{primordial}} \approx 0.76 \\ Y_{\text{primordial}} \approx 0.24 \\ Z_{\text{primordial}} \approx 0.00 \end{array} \right. \rightarrow \left\{ \begin{array}{l} X_{\odot} \approx 0.74 \\ Y_{\odot} \approx 0.25 \\ Z_{\odot} \approx 0.01. \end{array} \right. \qquad (\text{IV.12}) $$

The measure of metallicity is not completely straightforward and is vigorously debated (e.g. Kewley et al., 2019, for a review). In external galaxies, it is usually estimated by modeling observations of nebular optical lines, coming from HII regions. Several methods, allowing an observer to convert a few line ratios into a metallicity estimate, have been proposed. These methods have been calibrated on particular HII regions, modeling the photoionization, making unavoidable assumptions about the stellar populations and the topology of the gas. We have systematically compared several calibrations over the DustPedia sample (De Vis et al., 2019). We have favored the “S” calibration from Pilyugin & Grebel (2016), as it is the most reliable down to low metallicities.

IV.1.2.2 Production of Stardust

At the scale of a galaxy, the two most important sources of stardust are: (i) AGB stars, encompassing both LIMS winds and PNe; and (ii) SN II.

**AGB stars.** Most of the dust production in LIMS is believed to occur during the *Thermally-Pulsing Asymptotic Giant Branch* (TPAGB) phase (Gail et al., 2009). In addition, LIMS with $m_{\star} \lesssim 1 M_{\odot}$ do not condense grains (e.g. Ferrarotti & Gail, 2006). Theoretical models concur that only a fraction of the available heavy elements will go into stardust (Morgan & Edmunds, 2003; Ventura et al., 2012):

$$ \delta_{\text{LIMS}} \equiv \frac{m_{\text{stardust}}^{\text{ej}}}{m_{Z}^{\text{ej}}} \approx 10 - 40\%, \qquad (\text{IV.13}) $$

$m_{\text{stardust}}^{\text{ej}}$ and $m_{Z}^{\text{ej}}$ being the ejected mass of stardust and heavy elements. Observations and modeling of the circumstellar envelopes of post-AGB stars are consistent with these values (e.g. Ladjal et al., 2010).

**SN II.** There is solid evidence that grains form in *SuperNova Remnants* (SNR), as the ejected gas cools down. Theoretical estimates of the net dust yield of a single SN II range in the literature from $Y_{SN} \approx 10^{-3}$ to $Y_{SN} \approx 1 M_{\odot}/\text{SN}$ (e.g. Todini & Ferrara, 2001; Ercolano et al., 2007; Bianchi & Schneider, 2007; Bocchio et al., 2016; Marassi et al., 2019). From an observational point of view, measuring the dust mass produced *in situ* by a single SN II is quite difficult, as it implies disentangling the freshly-formed dust from the surrounding ISM. It also carries the usual uncertainty about dust optical properties. A decade ago, the largest dust yield ever measured was $Y_{SN} \approx 0.02 M_{\odot}$ (in SN 2003gd; Sugerman et al., 2006). The *Herschel* space telescope has been instrumental in estimating the cold mass of SNRs. The yields of the three most well-studied SNRs are now an order of magnitude higher:

**Cassiopeia A:** $Y_{SN} \approx 0.04 - 1.1 M_{\odot}$ (Barlow et al., 2010; Arendt et al., 2014; De Looze et al., 2017; Bevan et al., 2017; Priestley et al., 2019);

**The Crab nebula:** $Y_{SN} \approx 0.03 - 0.23 M_{\odot}$ (Gomez et al., 2012; Temim & Dwek, 2013; De Looze et al., 2019);

**SN 1987A:** $Y_{SN} \approx 0.45 - 0.8 M_{\odot}$ (Dwek & Arendt, 2015; Matsuura et al., 2015).

Most of the controversy however lies in the fact that, while large amounts could form in SN II ejecta (e.g. Matsuura et al., 2015; Temim et al., 2017), a large fraction of freshly formed grains would not survive the reverse shock ($v \approx 1000$ km/s; e.g. Nozawa et al., 2006; Micelotta et al., 2016; Kirchschlager
<figure><img src="image_425.png" /></figure>

et al., 2019). In all the cases we have listed above, the newly-formed grains have indeed not yet experienced the reverse shock (Bocchio et al., 2016). The net yield is thus expected to be significantly lower. Even if ≈ 10 − 20% of the dust condensed in an SN II ejecta survives its reverse shock (e.g. Nozawa et al., 2006; Micelotta et al., 2016; Bocchio et al., 2016), we have to also consider the fact that massive stars are born in clusters. The freshly-formed dust injected by a particular SN II, having survived the reverse shock, will thus be exposed to the forward shock waves of nearby SNe (e.g. Martínez-González et al., 2018). Overall, SN II dust yields are largely uncertain. We will extensively discuss their empirical constraint, from a statistical point of view, in Sect. IV.3. We will show that we can infer the average dust yield per SN II, 〈Y$_{SN}$〉. The corresponding timescale is then simply:

$$ \frac{1}{\tau_{\text{SN-cond}}(t)} = \frac{\langle Y_{\text{SN}} \rangle}{M_{\text{dust}}(t)} R_{\text{SN}}(t). \qquad (\text{IV.14}) $$

**Indirect evidence.** The best constraints on the fraction of ISD which is stardust might be indirect. The clear correlation between the depletion factor, F$_{⋆}$, and the average density of the ISM, 〈n$_{H}$〉, that we have discussed in Fig. II.17.a, has been shown to require rapid destruction and reformation into the ISM (e.g. Draine & Salpeter, 1979; Tielens, 1998; Draine, 2009). The rest of the grains needs to form in the ISM. Draine (2009, D09) gives a series of additional arguments concluding that, in the MW, stardust has to be less than 10% of ISD.

1. A first argument given by D09 is based on the typical lifetime of ISD.

a) From the literature, the stardust injection rate is roughly M$_{stardust}$ ≈ 5 × 10$^{-3}$ M$_{⊙}$/yr.

b) Noting that the typical lifetime of a dust grain in the MW is τ$_{ISD}$ ≈ 3 × 10$^{8}$ yr (cf. Sect. IV.2.2.3; Jones et al., 1996), the present stardust mass in the ISM should be: M$_{stardust}$ ≈ M$_{stardust}$ × τ$_{ISD}$ ≈ 1.5 × 10$^{6}$ M$_{⊙}$.

c) The ISD mass in the MW is roughly M$_{ISD}$ ≈ M$_{ISM}$/183 ≈ 2.7 × 10$^{7}$ M$_{⊙}$ (cf. Table II.4).

Stardust is thus only M$_{stardust}$/M$_{ISD}$ ≈ 5% of the total ISD.

2. A second argument given by D09 is based on the study of IDPs in meteorites (cf. Sect. II.2.4).

a) Stardust silicate grains in meteorites, identified with their isotopic anomalies (cf. Fig. II.19), appear to be ≈ 20% crystalline.

b) Silicates in the ISM are less than ≲ 2% crystalline (cf. Sect. II.2.1.2; Kemper et al., 2004).

Therefore, the fraction of stardust is only less than ≲ 2/20 ≈ 10% of ISD. Also, the fact that ISD is mainly amorphous, whereas circumstellar grains are essentially crystalline, is another argument in favor of rapid destruction and reformation in the ISM.

In the MW, stardust represents only a few percents of the ISD content.

# IV.2 Dust Evolution Processes in the ISM

Most of the important dust evolution processes occur in the ISM. These effects can be studied by looking at spatial variations of the dust properties in a region.

## IV.2.1 Grain Formation and Transformation

Grain formation is the transfer of elements from the gas phase to the dust phase, therefore increasing the dustiness.

<figure><img src="image_426.png" /></figure>
<figure><img src="image_427.png" /></figure>

IV.2.1.1 Evidence of Grain Growth and Coagulation in the ISM

We have just discussed stardust (cf. Sect. IV.1.2.2) which is thought to produce grain seeds onto which
mantle can grow. We now focus on the dominant process in Solar metallicity systems: the accretion
of gas phase atoms and molecules. Grain-grain coagulation does not result in grain formation per se,
as it does not affect the dustiness. It however follows grain growth and has similar effects on the FIR
opacity.

**The evidence brought by depletions.** As we have discussed in Sect. IV.1.2.2, the clearest evidence of grain growth in the ISM is provided by the good correlation between the depletion factor and the average density of the ISM (cf. Fig. II.17.a). It implies that atoms and molecules from the gas phase are progressively building up grain mantles, when going into denser regions. This observed behaviour is also consistent with the progressive de-mantling and disaggregation of cloud-formed, mantled and coagulated grains injected into the low density ISM, following cloud disruption. It is perhaps not unreasonable to hypothesise that dust growth in the ISM occurs on short timescales during cloud collapse rather than by dust growth in the quiescent diffuse ISM. In this alternative interpretation, the arrow of time is in the opposite sense and requires rapid dust growth, through accretion and coagulation, in dense molecular regions and slow de-mantling and disaggregation in the diffuse ISM (e.g. Jones, 2009). Given that astronomical observations provide only single-time snapshots, it will seemingly be difficult to determine the direction of the time-arrow of dust evolution.

**FIR opacity variations.** We have seen in Fig. I.21 that the growth of mantles has an impact on the FIR opacity (cf. Köhler et al., 2014, 2015). Yet, there is clear evidence of FIR opacity variations in the MW. The main factor seems to be the density of the medium. For instance, both Stepnik et al. (2003) and Roy et al. (2013) found that the FIR dust cross-section per H atom increases by a factor of ≈ 3 from the diffuse ISM to the molecular cloud they targeted. Stepnik et al. (2003) noticed that this opacity variation is accompanied by the disappearance of the small grain emission. They concluded that grain coagulation could explain these variations. In the diffuse ISM, Ysard et al. (2015) showed that the variation of emissivity, including the β − T relation (cf. Sect. III.1.2.1), could be explained by slight variations of the mantle thickness of the THEMIS model. For that reason, the THEMIS model aims at describing the evolution of grain mantles as a function of density and ISRF, as we have seen in Fig. I.21: (i) in the diffuse ISM, the grains are supposed to have a thin a-C mantle, largely dehydrogenated (aromatic) by UV photons; (ii) in denser regions, the mantle thickness is hypothesized to increase and to become more hydrogenated (aliphatic), because of the progressive shielding of stellar photons; (iii) in molecular clouds, grains are thought to be coagulated and iced. The THEMIS model predicts a factor of ≈ 2.5 dust mass increase in going from MW diffuse to dense clouds. This would correspond to a factor up to ≈ 7 in terms of dust emissivity per H atom (Köhler et al., 2015).

IV.2.1.2 Studies of the Magellanic Clouds

In nearby galaxies, studies of the local grain processing are difficult to conduct, as the emissivity vari-
ations are smoothed out by the mixing of dense and diffuse regions. Even when potential evolution-
ary trends are observed, their interpretation is often degenerate with other factors. The Magellanic
clouds are the most obvious systems where this type of study can be attempted. The insights provided
by depletion studies (cf. Sect. II.2.3) show that there are clear variations of the fraction of heavy ele-
ments locked-up in dust, and these variations correlate with the density (Tchernyshyov et al., 2015;
Jenkins & Wallerstein, 2017). Since the coagulation and the accretion of mantles lead to an increase
of FIR emissivity (cf. Fig. I.21; Köhler et al., 2015), we should expect emissivity variations in the Mag-
ellanic clouds. Indeed, Roman-Duval et al. (2017) studied the trends of gas surface density (derived
from H I and CO) as a function of dust surface density (derived from the IR emission), in these galax-
ies. They found that the observed dustiness of the LMC increases smoothly by a factor of ≈ 3 from the
diffuse to the dense regions. In the SMC, the same variation occurs, with a factor of ≈ 7. They argue

<figure><img src="image_428.png" /></figure>
that optically thick HⅠ and CO-free H₂ gas (cf. Sect. III.3.2.3) can not explain these trends, and that grain growth is thus the most likely explanation.

**Spatially-resolved SED fitting of LMC-N44 and SMC-N66.** We have conducted a similar study, focussing on two massive star-forming regions, rather than the whole galaxies<fnref n="5" />: (i) N44 in the LMC; and (ii) N66 in the SMC (Galliano, 2017). Our maps were 200 pc wide regions, with a spatial resolution of ≈ 15 pc. The MIR-to-submm data were coming from the SAGE/HERITAGE surveys (Spitzer and Herschel data; Meixner et al., 2006, 2013). We used the hierarchical Bayesian SED model, HerBIE, with the AC dust composition of Galliano et al. (2011, cf. Sect. III.1.3.3). The goal was to perform a spatially-resolved modeling of the dust properties, in a region with a strong gradient of physical conditions, in order to probe dust processing, as a function of density, ISRF and metallicity. The wide range of physical conditions can be estimated by looking at the range of SEDs shown in Fig. IV.8. In both panels, the faintest pixels show a rather cold SED, peaking around λ ≈ 100 µm, whereas the brightest pixels peak around λ ≈ 60 µm, with a very broad FIR bump, indicating a wide range of ISRF intensities, typical of compact SF regions (cf. Sect. III.1.2.2).

<figure><img src="image_429.png" /><figcaption>FIGURE IV.8 – Spatially-resolved SED fits in N44 and N66. In both panels, we show the brightest and the faintest pixels of our entire map. The grey error bars are the observations, and the colored densities are the Bayesian posterior distributions of SED models (Galliano, 2017). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Derived dust-gas relations.** We have compared the derived dust and total gas column densities. The latter was estimated from the [HⅠ]$_{21cm}$ and $^{12}$CO(J=1→0)$_{2.6mm}$ measurements (Meixner et al., 2006; Gordon et al., 2011; Meixner et al., 2013). The results are displayed in Fig. IV.9. The orange line represents the Galactic dustiness (cf. Table II.4) scaled by the metallicity, therefore representing the Galactic dust-to-metal mass ratio. This line corresponds to the values we would expect if the dust constitution was close to the diffuse ISM of the MW and was not evolving with density. The most diffuse pixels in both regions are consistent with this value. The hatched yellow area corresponds to a dustiness larger than the metallicity, that is requiring more heavy elements in dust than what is available in the ISM. Overall, the trends of Fig. IV.9 indicate a non-linear dust-to-gas relation, with a variation of the observed dustiness by a factor of ≈ 3, similar to the studies we have reviewed at the

<footnote n="5">5. As a reminder, the metallicities of the Magellanic clouds are: Z$_{LMC}$ ≈ Z$_{⊙}$/2 and Z$_{SMC}$ ≈ Z$_{⊙}$/5 (Pagel, 2003).</footnote>
<figure><img src="image_430.png" /></figure>

beginning of this section. The high density pixels lie in the forbidden zone. The possible causes are the following.

**Grain growth:** a part of this trend is likely the result of the evolution of the *true dustiness*, due to mantle accretion in denser regions. The yellow hatched area can however be considered as a hard upper limit, as in practice, not all heavy elements are refractory: even in the densest molecular clouds, there are gas phase CO, HCN, *etc*. This sole factor is therefore probably not sufficient to explain the full extent of the *observed dustiness* variation.

**Emissivity increase:** grain growth and coagulation are accompanied by an increase of FIR emissivity (cf. Fig. I.21; Köhler et al., 2015). This effect is naturally expected and would amplify the increase of the *observed dustiness* with density, as the constant emissivity assumption of our SED model would lead us to overestimate the dust mass of dense regions.

**Contribution of dark gas:** we have not accounted for CO-dark gas (cf. Sect. III.3.2.3). This component can potentially bias the molecular gas mass estimate in translucent regions by up to a factor ≈ 100 (Madden et al., 2020). It could also explain a part of the trend, as it would result in an underestimate of the total gas mass. This effect should however be significative at intermediate column densities, and decrease toward the densest regions, where CO would dominate. It might thus not be the main cause of the non-linearity of our trends.

<figure><img src="image_431.png" /></figure>

There is multiple evidence of dust evolution as a function of density, consistent with grain growth and coagulation, and the consequent increase of emissivity.

<figure><img src="image_432.png" /></figure>

**FIGURE IV.9** – *Dust-to-gas mass surface density relation in N44 and N66.* In both panels, we show the trends of dust mass surface density as a function of the total gas mass (atomic and molecular) surface density derived from the SED fitting of Fig. IV.8 (Galliano, 2017). Each point, with its uncertainty ellipse, corresponds to ≈ 15 pc pixel. The orange line corresponds to the dustiness of the MW (cf. Table II.4) scaled by the metallicity of the region. The yellow hatched area corresponds to a dustiness larger than the metallicity, that is an unrealistic value requiring more heavy elements being locked up in grains than what is available in the ISM. Licensed under CC BY-SA 4.0.

### IV.2.1.3 Quantifying Grain Growth

We now discuss the way grain growth can be approximately quantified. The following relations are rather uncertain, because of the lack of constraint on grain structure and composition. They however
<figure><img src="image_433.png" /></figure>

provide a framework to study grain growth efficiency.

**Accretion timescale.** Timescales for grains to accrete atoms are widely discussed in the literature (e.g. Dwek, 1998; Edmunds, 2001; Draine, 2009; Hirashita & Kuo, 2011; Zhukovska et al., 2016; Priestley et al., 2021). First, the collision rate of an atom E of mass $m_E$, with a grain of radius a is:

$$ \frac{1}{\tau_{\text{coll}}(a, E)} \equiv \underbrace{\pi a^2}_{\text{grain cross-section}} \times \underbrace{n_E}_{\text{gas density of E}} \times \underbrace{\sqrt{\frac{8kT_{\text{gas}}}{\pi m_E}}}_{\text{Maxwellian velocity of E}} . \qquad (\text{IV.15}) $$

In this equation, we have implicitly neglected Coulomb interaction (i.e. we have assumed that the grain and the atom are both neutral, which is a reasonable assumption in the CNM). Second, the growth rate of a grain of mass $m_d(a)$, due to accretion following these collisions, can be written:

$$ \left(\frac{dm_d}{dt}\right)_{\text{acc}} (a, E) = \underbrace{\mathcal{S}}_{\text{sticking probability}} \times \underbrace{\frac{m_E}{f_E}}_{\text{gained mass}} \times \underbrace{\frac{1}{\tau_{\text{coll}}(a, E)}}_{\text{rate}} \qquad (\text{IV.16}) $$

where $0 \le \mathcal{S} \lesssim 1$ is the *sticking coefficient*, that is the probability the atom will be bound with the grain after the collision. The factor $f_E$ is the mass fraction of element E within the grain. We choose E as a *key element* (Zhukovska et al., 2008), that is the element in the grain make-up that will have the longest collision time.

**For silicates**, the key element is Si, with $f_{Si} \approx 0.16$ for olivine and $f_{Si} \approx 0.24$ for pyroxene (assuming Fe:Mg=1:1). In other words, for each collision with Si, there are more collisions with O, Fe and Mg. Therefore, the dust mass gained between two Si collisions is the mass of a full crystal unit (SiO$_4$MgFe for olivine SiO$_3$Mg$_{0.5}$Fe$_{0.5}$ for pyroxene; cf. Sect. I.1.4.1).

**For carbon grains**, the mass is essentially C, as H is negligible. We thus have $f_C \approx 1$.

Finally, it is convenient to express this quantity as an *accretion timescale*, $\tau_{\text{acc}}(a)$:

$$ \frac{1}{\tau_{\text{acc}}(a)} \equiv \frac{1}{m_d(a)} \left(\frac{dm_d}{dt}\right)_{\text{acc}} (a, E) = \mathcal{S} \frac{\sqrt{m_E}}{f_E} \frac{3n_E}{2a\rho} \sqrt{\frac{2kT_{\text{gas}}}{\pi}}, \qquad (\text{IV.17}) $$

where we have simply developed $m_d(a) = 4/3\pi a^3\rho$ in the second equality, $\rho$ being the mass density of the grain. The density of the element E can be written as a function of the total H density, assuming its abundance scales with metallicity:

$$ n_E \approx \left(\frac{Z}{Z_{\odot}}\right) \left(\frac{E}{H}\right)_{\odot} n_H. \qquad (\text{IV.18}) $$

We therefore see that the grain growth timescale roughly obeys the following proportionality (assuming $\mathcal{S} = 1$ and olivine composition of silicates):

$$ \tau_{\text{acc}}(a) \approx \frac{100 \text{ cm}^{-3}}{n_H} \times \frac{Z_{\odot}}{Z} \times \sqrt{\frac{100 \text{ K}}{T_{\text{gas}}}} \times \frac{a}{100 \text{ nm}} \times \begin{cases} 57 \text{ Myr} & \text{for silicates} \\ 41 \text{ Myr} & \text{for carbon grains.} \end{cases} \qquad (\text{IV.19}) $$

As said above, these estimates are uncertain. We especially have no idea of the sticking probability, $\mathcal{S}$. Eq. (IV.19) however provides a description of the sensitivity of grain growth to density, size and metallicity. It is also indicative of the lower limit of these timescales.

<figure><img src="image_434.png" /></figure>
**Grain growth in different ISM phases.** Fig. IV.10 displays Eq. (IV.19) for carbon and silicate grains in the most relevant ISM phases. Timescales longer than the typical destruction timescales by SN II blast waves are irrelevant. That is the reason why this range is hatched in yellow in Fig. IV.10.

**In the WNM**, only the smallest grains ($a \lesssim 10$ nm) could grow. The WNM is thus not very suitable for grain growth.

**In the CNM**, all grain sizes can grow in less than 300 Myr. The CNM is thus a phase where dust growth could happen.

**In diffuse molecular clouds**, growth timescales are roughly similar to the CNM. The same conclusion therefore applies.

**In dense molecular clouds**, all relevant interstellar grain sizes can grow in less than $\approx 10$ Myr, which is also the typical lifetime of these clouds, in star-forming regions.

These timescales are consistent with the picture painted by the variation of elemental depletions across phases (cf. Fig. II.17.a). To estimate a global growth timescale, let's consider the radius corresponding to the average mass of the THEMIS model, in Table II.3:

$$ \sqrt[3]{\langle a^3 \rangle_a} \approx \begin{cases} 31 \text{ nm} & \text{for silicates} \\ 28 \text{ nm} & \text{for large a-C(:H).} \end{cases} \qquad (\text{IV.20}) $$

With these sizes, a typical accretion time in the CNM would be $\tau_{acc} \approx 58$ Myr for silicates, and $\tau_{acc} \approx 38$ Myr for large a-C(:H).

Grains can possibly grow in the CNM, on timescales of $\gtrsim 30-60$ Myr, and faster in molecular clouds.

<figure><img src="image_435.png" /><figcaption>FIGURE IV.10 – Grain growth timescales. Both panels represent Eq. (IV.19) for several of the phases in Table III.6 (taking $n_H = 10^4$ cm$^{-3}$ for the dense molecular phase). We have assumed $\mathcal{S} = 1$ for both silicates and carbon grains. We have highlighted timescales longer than the typical shock destruction timescale, $\tau_{SN-dest} \approx 300$ Myr (cf. Sect. IV.2.2.3), in hatched yellow. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Relation to global parameters.** As we will see in Sect. IV.3, it is convenient to relate the grain growth timescale to global galaxy parameters. Mattsson et al. (2012) proposed a relation based on the following assumptions.
1. Most of grain growth happens in molecular clouds. The mass surface density of these molecular clouds, $\Sigma_{mol}$, is proportional to the SFR surface density, $\Sigma_{SFR}$ (e.g. Kennicutt, 1998b). Grain growth rate is thus proportional to $\Sigma_{SFR}$:

$$ \frac{\Sigma_{\text{gas}}}{\tau_{\text{grow}}} \propto \Sigma_{\text{SFR}}. \qquad (\text{IV.21}) $$

2. The grain growth rate is also proportional to the fraction of available heavy elements in the gas. It implies that:

$$ \frac{\Sigma_{\text{gas}}}{\tau_{\text{grow}}} \propto \Sigma_{\text{SFR}} \times \left(1 - \frac{Z_{\text{dust}}}{Z}\right) Z, \qquad (\text{IV.22}) $$

where $Z_{dust}$ is the dustiness, and $Z_{dust}/Z$, the dust-to-metal mass ratio (cf. Sect. II.2.3.2). By subtracting $Z_{dust}/Z$, we account for the fact that the fraction of heavy elements already locked up in grains does not contribute to grain growth.

3. The other parameters in Eq. (IV.19) are assumed to not vary significantly. These parameters are the mean grain size, the mean gas velocity and mean density of molecular clouds.

The grain growth rate proposed by Mattsson et al. (2012) can thus be parametrized as a function of global galactic quantities and a phenomenological, dimensionless parameter, $\epsilon_{grow}$, containing all our uncertainties. The goal is to empirically infer $\epsilon_{grow}$, as we will see in Sect. IV.3. Eq. (IV.22) thus becomes:

$$ \frac{1}{\tau_{\text{grow}}(t)} = \epsilon_{\text{grow}} \frac{\psi(t)}{M_{\text{gas}}(t)} (Z(t) - Z_{\text{dust}}(t)), \qquad (\text{IV.23}) $$

where we have replaced the ratio of surface densities by the ratio of the quantities, and have explicit the temporal dependencies. In the case of the MW ($\psi \approx 1.3 M_{\odot}/\text{yr}$; $M_{gas} \approx 7 \times 10^9 M_{\odot}$), a grain growth timescale of $\tau_{grow} \approx 60 \text{ Myr}$ (Eq. IV.19) corresponds to $\epsilon_{grow} \approx 10^4$.

## IV.2.2 Grain Destruction

We now discuss grain destruction, that is the return of heavy elements from the grains to the gas phase. Note that fragmentation and shattering by shock waves (at $v \lesssim 200 \text{ km/s}$), that we have discussed in Sect. IV.2.2.3, simply rearrange the size distribution without destroying the dust. Shocks however have a pulverization effect, accompanying the other processes, that are difficult to differentiate from an observational point of view.

### IV.2.2.1 Photodestruction of Small Grains

Due to thermal spikes, small grains have a certain probability that one of their atom will be ejected. This is a runaway process leading to the complete sublimation of the dust grain.

**Photodesorption and sublimation.** Following the formalism of Guhathakurta & Draine (1989), we consider a cluster $X_N$ containing N atoms of X (X can be C, Fe, Si, O, etc.). The ejection of an atom from the grain is balanced by the return of an atom from the gas phase. The rate of the reaction $X_N + X \rightleftharpoons X_{N+1}$ is $R_N A_N$, where the total grain surface is $A_N = 4\sigma_N \approx 4\pi a^2$. Guhathakurta & Draine (1989) write the sublimation rate as:

$$ \frac{dN}{dt} = -R_N(T) \times S_N(T) \times A_N, \qquad (\text{IV.24}) $$

and provide the following rates for graphite and silicate:

$$ R_{N+1}^{\text{gra}}(T) \approx 4.6 \times 10^{33} \left(\frac{\alpha_N}{0.1}\right) \exp\left(-\frac{B_{\text{gra}}}{kT}\right) m^{-2} s^{-1}, \qquad (\text{IV.25}) $$

$$ R_{N+1}^{\text{sil}}(T) \approx 4.9 \times 10^{34} \left(\frac{\alpha_N}{0.1}\right) \exp\left(-\frac{B_{\text{sil}}}{kT}\right) m^{-2} s^{-1}, \qquad (\text{IV.26}) $$
<figure><img src="image_436.png" /></figure>

with the binding energy per atom:

$$B_{\text{gra}}/k = 81200 - 20000N^{-1/3} \text{ K}, \qquad (\text{IV.27})$$

$$B_{\text{sil}}/k = 68100 - 20000N^{-1/3} \text{ K}. \qquad (\text{IV.28})$$

The sticking coefficients, $\alpha_N$, is unknown and is arbitrarily chosen by the authors to be $\alpha_N \approx 0.1$. Assuming the surface free energy is about $2 \times 10^4$ K, the term $-20000N^{-1/3}$ accounts for the surface tension, making it easier to release an atom when the grain is smaller. Finally, the suppression factor, $S_N(T) < 1$, accounts for the suppression of the thermal fluctuations in a thermally isolated particle. This factor is:

$$S_N(T) = \left(\frac{1+\gamma}{\gamma}\right)^b \frac{\Gamma(\gamma f + 1)\Gamma(\gamma f + f - b)}{\Gamma(\gamma f - b + 1)\Gamma(\gamma f + f)}, \qquad (\text{IV.29})$$

where $f = 3N-6$ is the number of vibrational degrees of freedom (cf. Sect. I.2.3). In addition, the mean number of quanta per degree of freedom is $\gamma = H(T)/(\hbar\omega_0 f)$, and the number of quanta necessary to release a particle is $b = B/(\hbar\omega_0)$. Guhathakurta & Draine (1989) take $\hbar\omega_0 = 0.75\Theta$, where $\Theta$ is the Debye temperature (taking $\Theta_{\text{gra}} = 420$ K and $\Theta_{\text{sil}} = 470$ K; cf. Sect. I.2.3.2). The mean lifetime is then integrated over the temperature distribution:

$$\frac{1}{\tau_{\text{subl}}} = \int \frac{dP}{dT} R_N(T) S_N(T) A_N dT. \qquad (\text{IV.30})$$

Guhathakurta & Draine (1989) assume that a grain does not survive if it has a lifetime $\lesssim 10^{13}$ s $\approx 0.3$ Myr. Fig. IV.11 displays these lifetimes for silicates and graphite bathed in the Mathis et al. (1983) ISRF. Although the exact numbers are to be taken with caution, we can conclude the following.

1. In the diffuse ISM of the MW (U = 1), silicates larger than $a \approx 4.5$ Å and graphite $a \approx 3.5$ Å can survive.

2. When $a$ increases, the lifetimes become exponentially longer, meaning that for grains larger than $a \gtrsim 6$ Å, the survival of these grains will not be very sensitive to the assumptions in Eq. (IV.30).

The hardness of the ISRF, that we have not represented here, will however increase the minimum size a grain needs to have in order to survive. The vicinity of OB associations will thus be environments where the smallest grains can be photodestroyed.

**Evidence in resolved regions.** This last point is observationally verified in countless regions. It can be conveniently witnessed, as the smallest grains are the carriers of the MIR continuum, which is well separated from the rest of the emission (cf. Fig. II.27.b). In addition, small carbon grains carry the series of aromatic features (cf. Sect. III.2.1.1). The disappearance of these features in regions of enhanced ISRF is very likely the sign of the destruction of these grains by hard UV photons. This is, for instance, evident in one of our studies of the massive star-forming region, N11, in the LMC (Galametz et al., 2016). This region contains several blobs, with embedded star clusters. The maps of the PAH mass fraction, $q_{\text{PAH}}$ (cf. Sect. III.1.2.2), is shown in Fig. IV.12.a. It has been derived by modeling the spatially-resolved SED of Spitzer and Herschel images. Comparing this image to the mean starlight intensity in Fig. IV.12.b, we see that PAHs are strongly depleted in the blobs where $\langle U \rangle$ is enhanced. The photodestruction is evident. In the case of this massive region, a bright star cluster such as N11B can clear PAHs out over a region of typically $\approx 50$ pc.

Aromatic features are severely depleted around star-forming regions.

### IV.2.2.2 Thermal Sputtering

Another destruction mechanism is grain erosion and vaporization by collisions with energetic ions, either in coronal plasmas (thermal sputtering; $T \gtrsim 10^6$ K), or in shock waves (kinetic sputtering; $v \gtrsim$

<figure><img src="image_437.png" /></figure>
<figure><img src="image_438.png" /></figure>

<figure><img src="image_439.png" /><figcaption>FIGURE IV.11 – *Lifetimes of small grains in a radiation field.* In both panels, we plot the mean sublimation times from Eq. (IV.30), for different grain radii (color lines), as a function of the starlight intensity of the Mathis et al. (1983) ISRF: (a) for the silicates of Weingartner & Draine (2001a); and (b) for the graphite of Laor & Draine (1993). Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_440.png" /><figcaption>FIGURE IV.12 – *Carving out of PAHs by UV photons in N11.* Both maps have identical fields of view, centered on the star cluster N11B (LH10; Lucke & Hodge, 1970), and are 400 pc wide. The quantities displayed are the results from the spatially-resolved SED modeling, with the composite approach (cf. Sect. III.1.2.2; Galametz et al., 2016): (a) the PAH mass fraction, $q_{PAH}$; and (b) the mean starlight intensity, $\langle U \rangle$, in units of $2.2 \times 10^{-5}$ W/m$^2$. The black areas have been masked. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_441.png" /></figure>
100 km/s). There is an abundant literature on the subject (e.g. Draine & Salpeter, 1979; Dwek & Scalo, 1980; Tielens et al., 1994; Jones et al., 1996; Jones, 2004; Nozawa et al., 2006; Micelotta et al., 2010; Bocchio et al., 2012, 2014; Hu et al., 2019, see also the review by Dwek & Arendt, 1992). We start by discussing thermal sputtering in this section, and will review kinetic sputtering in Sect. IV.2.2.3.

**Sputtering times.** The evolution of a grain of radius, a, and mass, $m_d(a) = 4/3\pi a^3\rho$, subjected to sputtering in a gas of density $n_H$, can be expressed (e.g. Hu et al., 2019):

$$ \frac{dm_d(a)}{dt} = 3m_d(a) \frac{1}{a} \frac{da}{dt} = 3 \frac{m_d(a)}{a} n_H Y_{\text{sput}}(T_{\text{gas}}, v_s), \qquad (\text{IV.31}) $$

where we have hidden all the microphysics into the sputtering yield, $Y_{\text{sput}}(T_{\text{gas}}, v_s) \equiv da/dt/n_H$. This quantity depends on: (i) the gas temperature, $T_{\text{gas}}$, in case of thermal sputtering; or (ii) the shock velocity, $v_s$, in case of kinetic sputtering. A detailed derivation of $Y_{\text{sput}}$ can be found in Nozawa et al. (2006, Sect. 5). For our simple discussion, we will adopt their yields, for silicate and carbon grains, fitted by Hu et al. (2019). In the thermal case, the sputtering rate can be expressed as:

$$ \frac{1}{\tau_{\text{sput}}^{\text{th}}(a, T_{\text{gas}}, n_H)} \equiv \frac{1}{m_d} \frac{dm_d}{dt} = \frac{3n_H}{a} Y_{\text{sput}}(T_{\text{gas}}). \qquad (\text{IV.32}) $$

Fig. IV.13.a shows the lifetimes of grains in a coronal plasma. With Eq. (IV.32), in the HIM (cf. Table III.6), typical grains (Eq. IV.20) have lifetimes of: (i) $\tau_{\text{sput}}^{\text{th}} \approx 11$ Myr, for silicates; and (ii) $\tau_{\text{sput}}^{\text{th}} \approx 16$ Myr, for carbon grains. In the case of SN II blastwaves, grains stay in post-shock conditions for only $\approx 10^4$ yr. Dust destruction by thermal sputtering is thus not the dominant process in the shocked ISM.

Grains have short lifetimes in coronal plasmas.

<figure><img src="image_442.png" /><figcaption>FIGURE IV.13 – Thermal and kinetic sputtering times of silicates and carbon grains. Panel (a) show the mean lifetimes of grains in a hot gas, computed from Eq. (IV.32), using the sputtering yields of Nozawa et al. (2006) fitted by Hu et al. (2019). Panel (b) show the mean lifetimes of grains in a shock of velocity $v_s$, computed from Eq. (IV.33), using the same sputtering yields. Licensed under CC BY-SA 4.0.</figcaption></figure>