<figure><img src="image_162.png" /></figure>

<figure><img src="image_163.png" /><figcaption>(a) Balloon crash</figcaption></figure>

<figure><img src="image_164.png" /><figcaption>(b) PILOT</figcaption></figure>

<figure><img src="image_165.png" /><figcaption>(c) SOFIA</figcaption></figure>

**FIGURE II.3 – Airborne observatories.** Panel (a) shows the 2010 crash site of the Nuclear Compton Telescope (NCT) in order to illustrate the challenges of such observation campaigns. Panel (b) shows an artist rendering of the PILOT balloon with a zoom on the gondola where the telescope is (Bernard et al., 2016). Panel (c) shows a picture of SOFIA in flight, with the telescope door open. Credit: (a) courtesy of NASA; (b) Bernard et al. (2016), with permission from Jean-Philippe BERNARD; (c) courtesy of NASA.

1997-2010; $\emptyset$ = 2 m; Pascale et al., 2008); the PROjet National pour l'Observation Submillimétrique (PRONAOS; 1994-1999; $\emptyset$ = 2 m; Serra et al., 2002); the Polarized Instrument for the Long-wavelength Observation of the Tenuous ISM (PILOT; 2015-2017; $\emptyset$ = 0.73 m; Bernard et al., 2016, Fig. II.3.b).

**Airplanes.** Airplanes can fly up to $\approx$ 15 km altitude and operate during $\approx$ 10 h. Numerous flights can be scheduled, contrary to balloons, which usually do only a few flights in their whole lifetime. The telescope motion being limited by its orientation perpendicular to the plane (Fig. II.3.c), the flight path has to be adapted to the observed source. The *Kuiper Airborne Observatory* (KAO; 1974-1995; $\emptyset$ = 0.9 m; Erickson & Meyer, 2013) was a transport jet plane converted into an observatory. The *Stratospheric Observatory for Infrared Astronomy* (SOFIA; 2010-; $\emptyset$ = 2.5 m; Young et al., 2012, Fig. II.3.c) is its current successor. It is a retired Boeing 747, modified to host the telescope.

### II.1.1.4 Space Telescopes

**IRAS.** The *InfraRed Astronomical Satellite* (IRAS; 1983; $\emptyset$ = 0.57 m; Neugebauer et al., 1984) was the first observatory to perform an all-sky survey at IR wavelengths. It mapped the sky in four broad-bands centered at $\lambda$ = 12, 25, 60 and 100 µm, with angular resolutions of 0.5' – 2'. It opened the IR window, which was largely unexplored at the time. It discovered more than 300 000 point sources, many of them being starburst galaxies. These new objects, with deeply embedded star formation at the scale of the whole galaxy, emitting more than 95% of their luminosity in the IR, were unexpected (e.g. Soifer et al., 1987, for a review). The new categories of *Luminous InfraRed Galaxies* (LIRG) and *UltraLuminous InfraRed Galaxies* (ULIRG) were created to describe what had been observed. Dusty disks around stars were also discovered (Beichman, 1987). By accessing the cold grain emission, the first reliable dust masses of galaxies and Galactic clouds could be estimated. The IR emission provided a new constraint that shaped modern dust models (Désert et al., 1990). IRAS data are still used nowadays (e.g. Galliano et al., 2021, hereafter G21).

**COBE.** The *COsmic Background Explorer* (COBE; 1989-1993; $\emptyset$ = 0.2 m; Boggess et al., 1992) was aimed at mapping the CMB, as its name indicates. However, two of its three instruments were used to map the whole sky in the MIR and FIR, providing the main constraints on the emission of dust models until the *Planck* mission (Sodroski et al., 1994; Dwek et al., 1997). The third instrument, covering the microwave range was also instrumental in providing the first evidence of spinning grains.

<figure><img src="image_166.png" /></figure>
<figure><img src="image_167.png" /></figure>

**DIRBE** (*Diffuse Infrared Background Experiment*) was an instrument observing through ten broad-bands at: $\lambda = 1.25, 2.2, 3.5, 4.9, 12, 25, 60, 100, 140$ and $240$ µm (Hauser et al., 1998).

**FIRAS** (*Far-InfraRed Absolute Spectrophotometer*) was a low spectral resolution spectro-imager, observing between $\lambda = 100$ µm and 10 mm (Mather et al., 1999). At long wavelengths, the angular resolution was only $7^\circ$.

**DMR** (*Differential Microwave Radiometer*) was mapping the sky in three broadbands centered at $\lambda = 3.3, 5.6$ and $9.5$ cm (Smoot et al., 1994).

**ISO.** The *Infrared Space Observatory* (ISO; 1995-1998; $\emptyset = 0.6$ m; Kessler et al., 1996) was the first mission to extensively perform spectroscopy over the whole IR range. For that reason, it provided a wealth of data about all spectral features: silicates (Molster & Kemper, 2005), PAHs (Abergel et al., 2005; Sauvage et al., 2005), ices (Dartois et al., 2005). Studies of IR gas lines also took off: molecular (Habart et al., 2005) and ionized (Peeters et al., 2005). Finally, it refined our knowledge, through dust tracers, of star formation at all scales (Nisini et al., 2005; Verma et al., 2005; Elbaz et al., 2005). There were four instruments onboard.

**ISOCAM** (Cesarsky et al., 1996a) was a low spectral resolution MIR spectro-imaging camera, in the $\lambda = 2.5-17$ µm range. It also had twenty broad and narrow bands in the same range.

**SWS** (de Graauw et al., 1996) was a medium to high spectral resolution ($R \equiv \lambda/\Delta\lambda = 1000-35000$) MIR spectrometer in the $\lambda = 2.4-45$ µm range.

**LWS** (Clegg et al., 1996) was a low to medium spectral resolution ($R \approx 150-9700$) FIR spectrometer, in the $\lambda = 43-198$ µm range. Combined together, SWS and LWS provided several continuous spectra over the whole IR range ($\lambda = 2.4-198$ µm; e.g. Peeters et al., 2002b), that have never been equaled.

**ISOPHOT** (Lemke et al., 1996) was a photometer observing through several broad and narrow band filters, in the $\lambda = 2.5-240$ µm.

**Spitzer.** The *Spitzer space telescope* (cryogenic operation: 2003-2009; $\emptyset = 0.85$ m; Werner et al., 2004) was the successor of ISO. Its larger mirror size and more modern detectors allowed it to refine our understanding of what ISO discovered, and observe a significantly larger number of targets. Its angular resolution was $40''$ at $160$ µm. It had three instruments onboard.

**IRAC** (*InfraRed Array Camera*; Fazio et al., 2004) was performing photometry through four broad-bands, centered at $\lambda = 3.6, 4.5, 5.8$ and $8.0$ µm.

**IRS** (*InfraRed Spectrograph*; Houck et al., 2004) was a medium and high spectral resolution ($R = 90-600$) spectro-imager, observing in the $\lambda = 5.3-38$ µm range.

**MIPS** (*Multiband Imaging Photometer for Spitzer*; Rieke et al., 2004) was a photometer observing through three broadbands centered at $\lambda = 24, 70$ and $160$ µm.

**AKARI.** The *AKARI space telescope* (cryogenic phase: 2006-2008; $\emptyset = 0.69$ m; Murakami et al., 2007) was comparable to *Spitzer*. One of its advantages was its ability to record spectra down to $2$ µm, while *Spitzer/IRS* was limited to $5$ µm. AKARI performed an all-sky survey in several MIR to FIR bands. It had two instruments onboard.

**IRC** (*InfraRed Camera*; Onaka et al., 2007) was a MIR camera with numerous broad and narrow bands, as well as a low/mid-spectral resolution spectrometer, observing in the $\lambda = 1.8-26.5$ µm range.

**FIS** (*Far-Infrared Surveyor*; Kawada et al., 2007) was a FIR photometer observing through four broad-bands centered at $\lambda = 65, 90, 140$ and $160$ µm. It also had a *Fourier Transform Spectrometer* (FTS) over the same range.

<figure><img src="image_168.png" /></figure>
<figure><img src="image_169.png" /></figure>

**WISE.** The Wide-field Infrared Survey Explorer (WISE; 2009-2011; $\emptyset = 0.4$ m; Wright et al., 2010) was a MIR all sky surveyor. It mapped the sky through four broad photometric bands centered at $\lambda = 3.4, 4.6, 12$ and $22$ µm.

**Herschel.** The Herschel space observatory (2009-2013; $\emptyset = 3.5$ m; Pilbratt et al., 2010) was a FIR-submm mission. Its large mirror allowed it to reach subarcminute angular resolution at long wavelength (36" at $\lambda = 500$ µm). Combined with Spitzer data at shorter wavelengths, it gives access to the full dust emission and provides the most reliable dust property estimates of galaxies and Galactic regions. Herschel data allowed us to build large databases of galaxy dust properties (e.g. Davies et al., 2017). It also allowed us to better constrain the submillimeter grain opacity (Meixner et al., 2010; Galliano et al., 2011). Among its discoveries, it demonstrated the filamentary nature of star-forming regions (André et al., 2010). It had three instruments onboard.

**PACS** (*Photodetector Array Camera and Spectrometer*; Poglitsch et al., 2010) was an imager observing through three broadbands centered at $\lambda = 70, 100$ and $160$ µm. It also had a spectrometer that could target specific FIR lines.

**SPIRE** (*Spectral and Photometric Imaging REceiver*; Griffin et al., 2010) was a photometer observing through three broadbands centered at $\lambda = 250, 350$ and $500$ µm. It also had a FTS providing a continuous, medium spectral resolution spectrum over the $\lambda = 194-671$ µm spectral range.

**HIFI** (*Heterodyne Instrument for the Far-Infrared*; de Graauw et al., 2010) was a very high spectral resolution ($R \approx 10^7$) spectrometer covering the $\lambda = 157-625$ µm spectral range. It was designed to accurately measure gas line intensities.

**Planck.** The Planck space observatory (2009-2013; $\emptyset = 1.5$ m; Tauber et al., 2010) was a FIR-to-microwave satellite designed to study the cosmological background. It is a successor to COBE. It was launched in the same rocket as Herschel. It performed an all sky survey in all its bands (Fig. II.4). Planck had a larger beam than Herschel (5' at $\lambda = 1$ mm), but had an accurate absolute calibration. Its measure of the emission of the diffuse ISM of the MW is now the main constraint on dust models (e.g. Compiègne et al., 2011). Planck could also measure the linear polarization in all its bands. It thus provided unique constraints on the grain properties (e.g. Guillet et al., 2018) and maps of the Galactic magnetic field (e.g. Planck Collaboration et al., 2016b). It had two instruments onboard.

**HFI** (*High Frequency Instrument*; Lamarre et al., 2010) was a photometer/polarizer observing through six broadbands centered at $\lambda = 350, 550, 850, 1380, 2096$ and $2997$ µm.

**LFI** (*Low Frequency Instrument*; Mandolesi et al., 2010) was a photometer/polarizer observing through three broadbands centered at $\lambda = 4.3, 6.8$ and $10$ cm.

**The JWST.** The James Webb Space Telescope (JWST; 2021-; $\emptyset = 6.5$ m; McElwain et al., 2020) should be launched a few months after the time this manuscript is being written. Its large segmented mirror, that will unfold in space, will allow us to access sub-arcsec resolution in the MIR. It will have four instruments onboard.

**MIRI** (*Mid-InfraRed Instrument*; $\lambda = 5-27$ µm; Rieke et al., 2015) contains an camera and an imaging spectrometer. It will be the most relevant instrument to ISD studies.

**NIRspec** (*Near-InfraRed Spectrograph*; $\lambda = 0.6-5$ µm; Birkmann et al., 2016) is a NIR spectrometer.

**NIRcam** (*Near-InfraRed Camera*; $\lambda = 0.6-5$ µm; Beichman et al., 2012) is a NIR camera.

**NIRISS** (*Near-InfraRed Imager and Slitless Spectrograph*; $\lambda = 0.8-5$µm; Doyon et al., 2012) will perform NIR imaging and spectroscopy.

<figure><img src="image_170.png" /></figure>
<figure><img src="image_171.png" /><figcaption>(a) Uranography of all sky map projections</figcaption></figure>

<figure><img src="image_172.png" /><figcaption>(b) *Planck* all sky map of the dust polarized emission</figcaption></figure>

**FIGURE II.4** – *All sky maps.* Panel (a) describes the main features of the usual all sky projections. It shows both Galactic and extragalactic structures. Panel (b) shows the all sky map of the dust polarized emission. It is a *Van Gogh representation*, where the brush lines show the orientation of the projected magnetic field. Credit: (a) Jarrett (2004), with permission from Tom JARRETT; (b) copyright ESA/Planck collaboration, credit Marc-Antoine MIVILLE-DESCHÊNES, with his permission.
**UV and X-ray Satellites.** The UV spectral shape of the extinction curve is an important constraint on dust models. UV satellite have one advantage over IR instruments: they do not need to be cooled down. IR instruments indeed need a cryostat to limit their proper emission. The lifetime of IR missions is thus the lifetime of their helium supply, typically only a few years, whereas UV telescopes can operate during several decades. The most important UV missions are the following.

**IUE** (*International Ultraviolet Explorer*; 1978-1996; $\emptyset = 0.45$ m; *Boggess et al., 1978*) was the first important UV mission, expanding our view on dust extinction at $\lambda = 115-320$ nm.

**HST** (*Hubble Space Telescope*; 1990-; $\emptyset = 2.4$ m; *Burrows et al., 1991*) can take spectra in the near-UV range.

**FUSE** (*Far Ultraviolet Spectroscopic Explorer*; 1999-2007; $\emptyset = 1.5$ m; *Moos et al., 2000*) extended the spectral coverage of IUE at $\lambda = 90.5-110.5$ nm.

As we will see in Sect. II.2.1.3, the X-ray regime can provide interesting constraints on the dust properties. The most important missions are: ROSAT (1990-1999; $\emptyset = 0.84$ m; $\lambda = 0.06-30$ nm; *Aschenbach, 1991*), XMM-Newton (1999-; $\emptyset = 0.7$ m; $\lambda = 0.1-12$ nm; *Jansen et al., 2001*) and Chandra (1999-; $\emptyset = 1.2$ m; $\lambda = 0.1-12$ nm; *Weisskopf et al., 2002*). The *Advanced Telescope for High Energy Astronomy* (ATHENA; ≈ 2030; *Wilms et al., 2014*) will revolutionize the field.

### II.1.1.5 Grain-collecting Spacecrafts

<figure><img src="image_173.png" /><figcaption>(a) Aerogel honeycomb matrix</figcaption></figure>

<figure><img src="image_174.png" /><figcaption>(b) Aerogel dust track</figcaption></figure>

<figure><img src="image_175.png" /><figcaption>(c) X-ray image of a grain</figcaption></figure>

**FIGURE II.5 – Analysis of the Stardust mission.** Panel (a) shows the honeycomb matrix of Stardust. Each array is filled with an ultralight silica aerogel. It is ≈ 1000 times less dense than glass. Dust grains arriving at several km/s are slowed down without being pulverized. Panel (b) shows the cone-shaped track of one of the grains captured in the aerogel. Panel (c) shows the X-ray image of one of the grains. The magenta part corresponds to olivine crystals, surrounded by non-crystalline magnesium silicate in green. Credit: (a) courtesy of NASA/JPL; (b) courtesy of NASA/JPL; (c) Anna Butterworth/UC Berkeley from STXM data, courtesy of Berkeley Lab.

Electromagnetic waves are not the only vectors of information about ISD we can get. The motion of the heliosphere relative to the local interstellar cloud creates an inflow of ISD through the Solar system (at 26 km/s; e.g. Krüger et al., 2019, for a review). Contrary to interplanetary grains, this interstellar flow is important at high ecliptic latitude, allowing us to discriminate grains from extrasolar origins. Several spacecrafts have collected actual interstellar grains, and analyzed them in situ or returned them to Earth.

**Ulysses** (1990-2009; *Bame et al., 1992*) was a spacecraft designed to analyze the Solar wind. It was the first mission to capture dust grains from interstellar origin.

**Galileo** (1989-2003; *Johnson et al., 1992*) was a spacecraft sent to study Jupiter and its satellites. It detected interstellar grains on its way.
<figure><img src="image_176.png" /></figure>

**Cassini** (1997-2017; Matson et al., 2002) was a spacecraft sent to study Saturn and its satellites. It embarked an instrument called the *Cosmic Dust Analyzer* (CDA), recording the size, speed, direction and chemical composition of interstellar grains. It identified thirty six of them, smaller than 200 nm (Altobelli et al., 2016). They appeared to be essentially Mg-rich silicates with iron inclusions.

**Stardust** (1999-2006; Brownlee et al., 2003) was a spacecraft that captured grains in a low-density aerogel, and returned them to Earth for laboratory analysis (Fig. II.5). It identified seven interstellar grains (Westphal et al., 2014). Those were Mg-rich silicates, with sizes ≳ 1 µm. Two of them had crystalline structures.

## II.1.2 Chronology of the Main Breakthroughs

In what follows, we present the main discoveries about ISD. We order the discussion by themes. Table II.1 puts all these breakthroughs in chronological order. This is a partial and incomplete review. We refer the reader to van de Hulst (1986), Dorschner (2003), Whittet (2003), Li & Greenberg (2003) and Li (2005), for more complete historical reviews.

### II.1.2.1 Obscuration and Dimming of Starlight

**The discovery of dark nebulae.** The first evidence of ISD came through the obscuration of visible starlight. There was a debate during the whole XIXth century about the reality of this obscuration.

* Sir William HERSCHEL, in his treaty on *The Construction of the Heavens* (Herschel, 1785), reported “*an opening or a hole*” in the Scorpius constellation. This hole was later identified as the *Ink Spot* nebula, B86 (Fig. II.6.a; cf. Steinicke, 2016, for the historical branching out of this discovery).

* Edward BARNARD's photographs of Ophiucus showed dark lanes through the nebula (Barnard, 1899). They were at the time interpreted by Agnes CLERKE, as “*glades and clearing*” in the stellar distribution (Clerke, 1903).

* Twenty years later, Edward BARNARD (Fig. II.7.a) realized these black patches were actually “*real, obscuring masses, most probably dark nebulae*” (Barnard, 1919).

**The reddening of starlight.** The selective extinction of starlight provided the first consensual evidence of ISD. This was realized at the beginning of the 1930's.

* Already in the middle of the XIXth century, Friedrich Georg Wilhelm STRUVE found that the stellar volume density is decreasing with distance from the Sun (Struve, 1847). This decrease could be explained by a ≈ 1 mag/kpc absorption by interstellar material. This is a factor of ≈ 2 from the actual value, which is wavelength-dependent.

* Dark nebulae and the extinction of starlight were both understood by Henry N. RUSSELL (Russell, 1922). In advance on his time, he noticed that “*in certain instances stars embedded in dense luminous nebulosity are abnormally red*”. He finally deduced that this “*obscuration of light in space, therefore, whether general or selective with respect to wave-length, will be produced mainly by dust particles a few millionths of an inch in diameter*” (i.e. ≈ 0.03 – 0.1 µm).

* The seminal study of Robert J. TRUMPLER (Fig. II.7.b; Trumpler, 1930) provided the first solid evidence of ISD. His study was based on the observation, at the Lick observatory, of 100 open clusters. He found that the *diameter distances* (assuming all clusters have the same diameter) are always smaller than the *photometric distances* (based on the stellar spectral types), and this discrepancy increases with distance (Fig. II.6.b). This allowed him to interpret stellar color excesses as selective extinction by fine dust particles. He inferred particles of average mass

<figure><img src="image_177.png" /></figure>
<figure><img src="image_178.png" /></figure>

$10^{-19}$ g or larger$^4$. At the same time, Schalén (1929, 1931) independently arrived at a similar conclusions, at the Uppsala observatory.

<figure><img src="image_179.png" /><figcaption>(a) The Ink Spot nebula</figcaption></figure>

<figure><img src="image_180.png" /><figcaption>(b) Trumpler (1930)'s relation</figcaption></figure>

FIGURE II.6 – *First evidences of interstellar dust.* Panel (a) shows the *Ink spot nebula* (B86), that was originally mistaken by Herschel as a “*hole in the heavens*”. Panel (b) shows the relation between the photometric and diameter distances of 100 open clusters (Fig. 1 of Trumpler, 1930). The non-linearity of this relation provided the first unambiguous evidence for ISD. Notice the absence of error bars: this plot is from a long gone epoch, when major scientific discoveries could be accompanied by a feeling of airiness and eyeball statistics. Credit: (a) Gábor Tóth Astrophotography, licensed under CC BY-NC-ND; (b) Trumpler (1930).

### II.1.2.2 The Dust Continuum

**The Shape of the extinction curve.** The investigation of the spectral shape of the extinction curve started right after Trumpler's study.

* A series of papers (Rudnick, 1936; Hall, 1937; Greenstein, 1938; Stebbins et al., 1939) concluded that $κ(λ)$ was roughly proportional to $1/λ$ in the $λ = 0.3–1 \ \mu m$ range. By the end of the 1930's, it was thus clear that the grains responsible for the visible light extinction were not in the Rayleigh regime ($κ_{sca}(λ) ∝ 1/λ^4$; Sect. I.2.2.3).

* Stebbins & Whitford (1943) extended the study of extinction curves to the $0.35 \ \mu m < λ < 1.03 \ \mu m$ range, and found deviations to the $1/λ$ behavior, due to the now well-known far-UV rise and NIR knee (Sect. II.2.1).

* During the 1950's and 1960's followed a discussion on the universality of the shape of the extinction curves (cf. references in Li, 2005).

* Starting in the 1960's, the first airborne and space observatories opened the UV window, up to the Lyman break (York et al., 1973). The 2175 Å bump was discovered by Stecher (1965, cf. Sect. II.1.2.3).

* The first UV extinction curves in the *Large Magellanic Cloud* (LMC) were measured by Borgman et al. (1975), and by Koornneef (1978) in 30 Doradus. In the *Small Magellanic Cloud* (SMC), it was first measured by Rocca-Volmerange et al. (1981). In both cases, the steepness of the curve and the weakness of the bump were noted.

<footnote n="4">4. Assuming $ρ ≈ 3 \ g/cm^3$, the radius of these particles would be $a ≈ 2$ nm or larger.</footnote>
* Cardelli, Clayton, & Mathis (1989) performed a mathematical fit to a collection of extinction curves toward different sightlines in the MW. They found that, over the 0.125 µm < λ < 3.5 µm range, the shape is universal and controlled by a single parameter, R(V) = A(V)/(A(B) – A(V)) (where A(λ) is the magnitude extinction; cf. Sect. II.2.1).

* ISO observations of the Galactic center exhibited a flatter MIR curve (Lutz et al., 1996), confirmed by Indebetouw et al. (2005) with Spitzer data.

**Polarization by dichroic extinction.** Hall (1949) and Hiltner (1949) found that starlight was linearly polarized.

* Davis & Greenstein (1951) proposed that this polarization is due to dichroic extinction by nonspherical particles aligned on the magnetic field (cf. Sect. I.2.2.5).

* Serkowski (1973) determined the wavelength dependence of the polarization fraction, known as the *Serkowski curve* (cf. Sect. II.2.1.4).

<figure><img src="image_181.png" /><figcaption>(a) Edward E. BARNARD
(1857–1923)</figcaption></figure>

<figure><img src="image_182.png" /><figcaption>(b) Robert J. TRUMPLER
(1886–1956)</figcaption></figure>

<figure><img src="image_183.png" /><figcaption>(c) Hendrik C. VAN DE HULST
(1918–2000)</figcaption></figure>

FIGURE II.7 – *The pioneers*. Credit: (a) Wikipedia, public domain; (b) Weaver & Weaver (1957); (c) Rob BOGAERTS, licensed under CC BY-SA 3.0 NL.

**Dust emission.** The thermal emission of heated grains started to be observed in the 1960's. The presence of very small grains or large molecules with $a \lesssim 1$ nm was speculated by Platt (1956, they are known as *Platt particles*).

* Greenberg (1968) first realized that small grains must be stochastically heated.

* Andriesse (1978) reported the first observational evidence of such transiently heated Platt particles in M 17. The spectral shape of the MIR spectrum was indeed constant over the region, and much wider than a single grey body emission, consistent with temperatures up to 150 K.

* Similarly, the NIR continuum and 3.3 µm feature emission of several reflection nebulae was shown by Sellgren et al. (1983) to be consistent with small grains fluctuating up to 1000 K.

* The presence of small grains in the diffuse ISM was clearly evidenced by the 12 and 25 µm IRAS emission (Boulanger & Perault, 1988). It was observed by numerous studies afterward in all types of interstellar regions and galaxies.

### II.1.2.3 Identification of Dust Features

The confirmation of the presence of various solid-state and molecular features was important to better constrain the dust composition.
**Silicates.** The first identification of silicates was reported by Woolf & Ney (1969), in absorption toward M giant and supergiant stars. Kemper et al. (2004) provided a 2% upper limit on the crystalline silicate fraction, based on ISO observations toward the Galactic center. The MIR features, proper to crystalline silicates were indeed not detected.

**Carbonaceous grains.** MIR aromatic emission features were first detected in the *Planetary Nebula* (PN) NGC 7027 by Gillett et al. (1973, cf. Fig. II.8.a). They were called at the time *Unidentified Infrared Bands* (UIBs). They were attributed to the bending and stretching modes of PAHs ten years later (Duley & Williams, 1981; Léger & Puget, 1984; Allamandola et al., 1985, cf. Fig. II.8.b). The 3.4 µm aliphatic feature in absorption was first detected toward the Galactic center by Willner et al. (1979).

<figure><img src="image_184.png" /></figure>

**FIGURE II.8** – *First detection of UIBs.* Panel (a) shows the first detection of the 8.6 and 11.3 µm aromatic features by Gillett et al. (1973), in the planetary nebulae NGC 7027 and BD+30°3639. The red wing of the 7.7 µm is also visible. Panel (b) shows the qualitative comparison between a laboratory spectrum of soot and the UIBs in the Orion bar, by Allamandola et al. (1985). Credit: (a) Fig. 1 of Gillett et al. (1973); (b) Fig. 1 of Allamandola et al. (1985).

**Ices.** Ice absorption features have been searched for since the 1940's and the *dirty ice model* (cf. Sect. II.1.2.5).

* The 3.1 µm H₂O ice band was finally detected by Gillett & Forrest (1973).

* Lacy et al. (1984) provided the first observational identification of CO ice absorption.

* CO₂ ice absorption was discovered in ISO/SWS spectra (de Graauw et al., 1996; D'Hendecourt et al., 1996; Guertler et al., 1996).

**X-ray edges.** The absorption of X-ray photons by inner electronic shells can provide information on the crystalline configuration of solids (e.g. Forrey et al., 1998; Draine, 2003c, for the theoretical predictions). The first X-ray absorption edges were detected in Chandra and XMM-Newton data (Paerels
<figure><img src="image_185.png" /></figure>

et al., 2001), but their interpretation remained problematic. More recent studies have been able to constrain grain structures using these features (e.g. Lee et al., 2009b).

### II.1.2.4 Dusty Epiphenomena

**Diffuse Interstellar Bands.** There are unidentified, ubiquitous absorption features in the $\lambda \approx 0.4 - 2$ µm range, called *Diffuse Interstellar Bands* (DIBs; cf. Sect. II.2.1.5).

* They were first reported by Heger (1922a,b).

* It was only Merrill (1934) who showed their interstellar nature.

* More than 400 of them have been identified toward a diversity of sightlines, even in external galaxies (Hobbs et al., 2009). They remain largely unidentified, although four of them have been attributed to $C_{60}^+$ (Campbell et al., 2015; Walker et al., 2015).

**Extended Red Emission.** The *Extended Red Emission* (ERE) is a broad emission band, found in the $\lambda \approx 0.6 - 0.9$ µm range of a diversity of Galactic environments. It is attributed to dust photoluminescence (e.g. Witt & Vijh, 2004), but the nature of its carriers is still debated. Photoluminescence is a non-thermal emission process in which, subsequently to the absorption of a UV photon, a grain is brought to an excited electronic state. After partial internal relaxation, a redder photon is emitted, bringing the electron back to its fundamental state. The ERE was first reported in the *Red Rectangle* reflection nebula (Schmidt et al., 1980).

**Spinning Grains.** The radio emission of fastly spinning dust grains was predicted by several authors (Erickson, 1957; Hoyle & Wickramasinghe, 1970; Ferrara & Dettmar, 1994).

* Kogut et al. (1996) detected an *Anomalous Microwave Emission* (AME) in the COBE/DMR data. This excess could not be explained by thermal dust, free-free and synchrotron emissions.

* Draine & Lazarian (1998a,b) showed the AME could be explained by the rotational emission of small, charged, fastly rotating grains.

### II.1.2.5 Dust Models

**First models.** The first dust models of the 1930's, following Trumpler's study, were mainly speculative.

* The first assumptions about the dust composition were made by analogy with micrometeorites. Schalén (1936) and Greenstein (1938) assumed grains were made of $a = 0.01$ µm iron particles. Lindblad (1935) had assumed dust could form by condensation in space.

* The *dirty ice model* (Oort & van de Hulst, 1946; van de Hulst, 1949) was the first attempt to base the dust composition on the abundant atoms: H, C, N, O. These atoms would form various ices ($H_2O$, $CH_4$, $NH_3$) nucleating on grain seeds. The predicted dust temperature of this model was $\approx 15$ K, close to the actual value ($\approx 18$ K).

* Several studies hypothesized that graphite could be a major dust constituent, explaining the polarization of starlight, because of its anisotropic crystalline structure (Cayrel & Schatzman, 1954; Hoyle & Wickramasinghe, 1962). Graphite was supported by the discovery of the 2175 Å bump (Stecher, 1965; Stecher & Donn, 1965).

* Donn (1968) hypothesized that PAHs could be responsible for the 2175 Å bump, which was supported by the laboratory measurements of Joblin et al. (1992).

* The *MRN model* (Mathis, Rumpl, & Nordsieck, 1977) was the first attempt at fitting the average extinction curve of the MW with a mixture of graphite and silicate grains, with a power-law size distribution:

$$f(a) \equiv \frac{dN}{da} \propto a^{-3.5} \quad \text{for } a_- < a < a_+ \qquad (II.1)$$

<figure><img src="image_186.png" /></figure>
<figure><img src="image_187.png" /><figcaption>FIGURE II.9 – *Désert et al. (1990) dust model.* These two panels show the DBP90 model extinction and emission, and how it fits the observations of the diffuse Galactic ISM. We have shown its three components: PAH, VSG and BG. Notice the vintage-looking square shaped PAH features of this pre-ISO model. Licensed under CC BY-SA 4.0.</figcaption></figure>

where $[a_-, a_+] = [0.005, 1] \mu m$ for graphite and $[a_-, a_+] = [0.025, 0.25] \mu m$ for silicates.

**Calculation of the optical properties.** Dust models rely on the computation of optical properties. The techniques have improved with time. The laboratory measurements on the most relevant compounds also expanded.

* Mie theory (cf. Sect. I.2.2.3) was independently developed by Gustav MIE and Peter DEBYE (Mie, 1908; Debye, 1909).

* The DDA method (cf. Sect. I.2.2.4) was developed by Purcell & Pennypacker (1973).

* Draine & Lee (1984) presented the first UV-to-mm optical properties of astronomical silicate and graphite. These properties have been refined by numerous studies afterward.

* Similarly, synthetic optical properties of a mixture of astronomical PAHs were presented by Li & Draine (2001) and updated by Draine & Li (2007).

* The properties of various amorphous carbon compounds were inferred from laboratory data by Rouleau & Martin (1991) and Zubko et al. (1996). Jones (2012a,b,c) presented a physical parametrization of the optical properties of a-C(:H).

**Elemental depletions.** Elemental depletions (cf. Sect. II.2.3) are an important set of constraints on the dust mass and on the stoichiometry of the dominant grain compounds. Greenberg (1974) laid the ground for such studies. Savage & Bohlin (1979) showed that the depletion strength correlates well with the average density of the gas. Several studies have refined this approach. Jenkins (2009) presented a unified view, showing depletions were controlled by a single parameter, correlated with the column density.

**Modern panchromatic models.** With the COBE and IRAS data, dust models started to have the possibility to be constrained by both the emission and the extinction. These simultaneous constraints are important to break the degeneracy between the composition and the size distribution.
* The Désert, Boulanger, & Puget (1990) model (hereafter DBP90) was the first silicate-carbon-PAH model, fitting the extinction and emission of the Galactic diffuse ISM (Fig. II.9). This model had three components: the PAHs, the Very Small Grains (VSG; small carbon grains) and the Big Grains (BG; large carbon-coated silicates).

* Dwek et al. (1997) built the first model constrained by the fine spectral sampling of the COBE data (DIRBE broadbands and FIRAS spectrum).

* Zubko, Dwek, & Arendt (2004) added the elemental depletions to the emission and extinction, to constrain more accurately the grain composition.

* Jones et al. (2013, 2017) developed the *The Heterogeneous Evolution Model for Interstellar Solids* (THEMIS). This model is physically parametrized to account for the known evolution of the dust mixture as a function of the physical conditions.

* Guillet et al. (2018) presented a dust model accounting for the polarized emission observed by Planck.

## II.2 The Current Empirical Constraints

We now review the current empirical constraints that are used to build dust models. These models are calibrated on observations of the diffuse Galactic ISM. This medium indeed presents several advantages.

* It is optically thin. No radiative transfer is thus needed.

* It appears to be rather uniformly illuminated. The mixing of heterogeneous physical conditions along the sightline is probably limited.

* Dust properties of the diffuse ISM appear rather uniform.

* It benefits from a wealth of observations.

Observations of the diffuse Galactic ISM are thus the most important ones. Extragalactic constraints are the focus of Chap. III. It is currently impossible to gather the same type of data set, at the same level of accuracy, in external galaxies.

Several reviews discuss the available dust observables (e.g. Draine, 2003a; Dwek, 2005; Draine, 2009; Galliano et al., 2018; Hensley & Draine, 2021). We have represented on Fig. II.10 most of these observables on top of the typical SED of a gas-rich galaxy.

☞ A fundamental local quantity of the ISM is the dust-to-gas mass ratio or dustiness<fnref n="5" />:

$$Z_{\text{dust}} \equiv \frac{M_{\text{dust}}}{M_{\text{gas}}}. \qquad (\text{II.2})$$

### II.2.1 Extinction

Dust extincts light from the X-rays to the MIR. The effect of dust extinction on a background source is sometimes referred to as reddening. It is indeed more important on the blue side of the visible window.

<footnote n="5">5. We are trying to promote the term *dustiness*, introduced by G21, as it is much more concise than *dust-to-gas mass ratio*.</footnote>
<figure><img src="image_188.png" /></figure>

THE PREHISTORY

1785 ✶ Herschel's Construction of Heavens

1800 ✶ Herschel's discovery of infrared radiation

1847 ✶ Struve's dimming of starlight

1880 ✶ First deep-sky photograph by Henry DRAPER

1900 ✶ Planck's black body radiation

1908 ✶ Mie theory

1919 ✶ Barnard's obscuration

1922 ✶ Heger's first observation of DIBs

THE CLASSICAL ERA

1930 ✶ Trumpler's color excess study

1934 ✶ Interstellar nature of DIBs

1936 ✶ Small metallic particle model

1949 ✶ Dirty ice model

1949 ✶ Polarization of starlight

1965 ✶ Discovery of the 2175 Å bump

1969 ✶ First observation of silicate features

1970 ✶ First dust radiative transfer models

1973 ✶ First detection of the UIBs

1973 ✶ Serkowski curve

1977 ✶ MRN model

1978 ✶ First evidence of small, stochastically heated grains

1979 ✶ First detection of the 3.4 µm feature

1980 ✶ First detection of ERE

1983 ✶ ISRF of Mathis, Mezger & Panagia

THE SPACE AGE

1983 ✶ Launch of IRAS

1984 ✶ Draine & Lee optical properties

1984 ✶ PAHs proposed to explain the UIBs

1989 ✶ Launch of COBE

1989 ✶ Parametrization of the Galactic extinction curve by Cardelli, Clayton & Mathis

1990 ✶ Désert, Boulanger & Puget model

1995 ✶ Launch of ISO

1996 ✶ First detection of AME

2003 ✶ Launch of Spitzer

2004 ✶ Zubko, Dwek & Arendt model

2009 ✶ Launch of Herschel & Planck

2011 ✶ Revision of dust opacities

2013 ✶ THEMIS model

2015 ✶ Whole-sky maps of the polarized dust emission

2015 ✶ Identification of two DIBs

2018 ✶ Polarized dust emission model

2021 ✶ Launch of JWST?

...

THE QUANTUM AGE?

<figure><img src="image_189.png" /></figure>
<figure><img src="image_190.png" /></figure>

<figure><img src="image_191.png" /><figcaption>FIGURE II.10 – *Panchromatic dust observables.* Spectral energy distribution of a typical late-type galaxy (Galliano et al., 2018). The blue hatched area shows the power absorbed by dust. Typical DIB, ERE and AME spectra are shown, together with the most relevant gas lines. The free-free continuum is emitted by the deceleration of free electrons scattering off ions in ionized regions. The synchrotron continuum is emitted by electrons spiraling through the magnetic field. Here, $L_V$ is the electromagnetic power emitted per unit frequency. Inset: Model D of Guillet et al. (2018), with $G_0 = 100$; $\tau$ is the optical depth, $p$ is the starlight polarization degree, $P$ is the polarization intensity, $I$ is the total intensity, $\vec{P}$ is the light polarization vector, and $\vec{B}$ is the magnetic field vector. Licensed under CC BY-SA 4.0.</figcaption></figure>

### II.2.1.1 UV-to-NIR Extinction

**The extinction magnitude.** We saw in Sect. II.1.2.1 that the first dust studies were performed in extinction, in the visible range. Consequently, extinction properties were characterized by quantities depending on the magnitude system. The magnitude, $m(\lambda_0)$, of a star of flux $F_V(\lambda_0)$, observed through a photometric filter centered at wavelength $\lambda_0$, is:

$$m(\lambda_0) = -2.5\log\left(\frac{F_V(\lambda_0)}{F_V^0(\lambda_0)}\right), \qquad (II.3)$$

where $F_V^0(\lambda_0)$ is the reference flux or zero-point of the photometric filter. The zero-point is a calibration quantity, independent of the observed source. Two important bands for characterizing extinction are the B and V bands, centered respectively at $\lambda_B = 0.44$ µm and $\lambda_V = 0.55$ µm (Table B.5). The total extinction or extinction in magnitude is defined as:

$$A(\lambda_0) = m_{\text{obs}}(\lambda_0) - m_{\text{int}}(\lambda_0) = 2.5 \log \left( \frac{F_V^{\text{int}}(\lambda_0)}{F_V^{\text{obs}}(\lambda_0)} \right), \qquad (II.4)$$

<figure><img src="image_192.png" /></figure>
<figure><img src="image_193.png" /></figure>

where the index “obs” refers to the observed quantity, and “int” refers to the intrinsic quantity, that is the quantity not affected by dust extinction. In the MW, the average V-band extinction over the distance to the star, d, is A(V)/d ≈ 1.8 kpc⁻¹ (e.g. Whittet, 2003). A(λ₀) can be linked to a more physical quantity, the optical depth:

$$ \tau(\lambda) = \underbrace{\kappa(\lambda)}_{\text{dust opacity}} \times \underbrace{\rho}_{\text{ISM density}} \times \underbrace{L}_{\text{length of the sightline}} = \kappa(\lambda) \times \underbrace{\frac{Z_{\text{dust}}}{1 - Y_{\odot} - Z_{\odot}}}_{\text{H mass fraction}} \times \underbrace{m_{\text{H}}}_{\text{H atom mass}} \times \underbrace{N_{\text{H}}}_{\text{column density}} . \quad (\text{II.5}) $$

The expression above has been derived assuming homogeneous properties along the sightline (cf. Sect. III.1.1 for a more rigorous definition of τ). The observed flux can conveniently be expressed as a function of the optical depth:

$$ F_{\nu}^{\text{obs}}(\lambda) = F_{\nu}^{\text{int}}(\lambda) \times \exp[-\tau(\lambda)]. \quad (\text{II.6}) $$

Eq. (II.4) therefore implies that: A(λ) = 1.086 × τ(λ).

**The selective extinction.** The spectral shape of the extinction curve varies among sightlines. It can be quantified by the selective extinction, E(λ - λ₀) ≡ A(λ) - A(λ₀). In the MW, Cardelli, Clayton, & Mathis (1989) showed that the UV-to-NIR extinction curves follow a universal law, parametrized by the sole visual-to-selective extinction ratio:

$$ R(V) = \frac{A(V)}{A(B) - A(V)}. \quad (\text{II.7}) $$

This parametrization is demonstrated in Fig. II.11.a. We see that A(V) is a scaling parameter quantifying the amount of extinction along the sightline. According to Eq. (II.5), A(V) ∝ Zdust × NH. In the MW, there are no drastic dustiness variations, thus A(V) ∝ NH. On the other hand, R(V) is a shape parameter. It typically varies between R(V) ≈ 2 and R(V) ≈ 5. On average, R(V) ≈ 3.1 in the MW. Curves with R(V) ≳ 3.1 tend to be flatter.

The amount of extinction in the MW is NH/E(B-V) ≈ 8.8 × 10²⁵ m⁻² mag⁻¹ (Lenz et al., 2017), or, for R(V) = 3.1, NH/A(V) ≈ 2.8 × 10²⁵ m⁻² mag⁻¹.

The most notable features of the UV-to-NIR extinction curves are the following (cf. Fig. II.11.a).

**The Far-UV (FUV) rise** is mainly due to the absorption by small grains, in the Rayleigh regime (A(λ) ∝ 1/λ; cf. Sect. I.2.2.3).

**The 2175 Å bump** is attributed to small sp²-hybridized C bonds (cf. Sect. I.1.4).

**The optical knee** is mainly due to scattering by larger grains.

**The NIR extinction** can be approximated by a power-law: A(λ) ∝ λ⁻⁶, with α ≈ 2.27 (Maíz Apellániz et al., 2020).

**Measuring extinction.** The original method to measure the wavelength dependence of the extinction curve is the pair method (Stecher, 1965): two stars of the same spectral type are observed, one with a low, and one with a high foreground extinction. The extinction curve is directly derived from the differential SED or spectrum, assuming the dust properties are uniform along both sightlines. An alternative to this method consists in replacing the reference star by a synthetic spectrum, knowing the precise spectral type of the star (e.g. Fitzpatrick & Massa, 2005). This is demonstrated in Fig. II.11.b.

**UV-visible scattering.** Observations of starlight scattering by ISD can constrain the average albedo, ω, and asymmetry parameter, ⟨cos θ⟩, of the grains (cf. Sect. I.2.2.3). Two types of observations are usually favored to that purpose.

<figure><img src="image_194.png" /></figure>
<figure><img src="image_195.png" /></figure>

FIGURE II.11 – Galactic extinction curves. Panel (a) represents the average extinction curves from the spectroscopic sample of Fitzpatrick et al. (2019), for different values of R(V). Panel (b): the blue line represents the synthetic, intrinsic UV-visible SED of a B star from the Lanz & Hubeny (2007) library; the red line is the extincted SED with A(V) = 1 and R(V) = 3.1. Licensed under CC BY-SA 4.0.

**The Diffuse Galactic Light (DGL)** is the scattering of the general ISRF by dust. It is the diffuse visible light seen in ISM regions without associated stars. The DGL was first detected by Elvey & Roach (1937). Henyey & Greenstein (1941) built their scattering phase function (cf. Sect. I.2.2.3) to explain this phenomenon.

**Reflection nebulae** are obvious objects to measure $\tilde{\omega}$ and $\langle\cos\theta\rangle$, as the visible light comes from a nearby star or cluster, and is scattered on the surface of a cloud facing us.

Both methods converge toward qualitatively consistent results:

☞ ISD has a UV-visible albedo around $\tilde{\omega} \approx 0.5$, and is rather forward scattering ($\langle\cos\theta\rangle \gtrsim 0.5$), meaning grains are not in the Rayleigh regime (cf. Sect. I.2.2.3).

### II.2.1.2 MIR Extinction

**The MIR continuum.** The spectral shape of the MIR extinction is harder to constrain than its UV-visible counterpart. The MIR range is indeed the domain where the stellar and dust SEDs intersect (cf. Fig. II.10). It is therefore difficult to precisely model the background sources toward which the extinction is measured.

* The Rayleigh-Jeans continuum of old stars, peaking in the NIR, such as G and K-type stars, can be used (e.g. Xue et al., 2016).

* Otherwise, H recombination lines (e.g. Lutz et al., 1996) or H$_2$ rovibrational lines (e.g. Bertoldi et al., 1999) provide an alternative. The ratio of a series of these lines can indeed be reasonably well modeled. Differences between the theoretical and observed ratios result from the extinction.

In the ISO days, there was a controversy about the 4-to-8 µm continuum, which seemed to be following the extrapolation of the NIR power-law trend (e.g. Bertoldi et al., 1999). However, Lutz et al. (1996) showed this continuum toward the Galactic center was relatively flat (cf. Fig. II.12.a). This has been confirmed by ulterior observations with Spitzer, WISE and AKARI (e.g. Indebetouw et al., 2005; Xue et al., 2016; Gordon et al., 2021). It seems to be a general feature of a wide variety of sightlines. The new synthetic extinction of Hensley & Draine (2021) reproduces this flat continuum (cf. Fig. II.12.a).
<figure><img src="image_196.png" /></figure>

<figure><img src="image_197.png" /><figcaption>**FIGURE II.12 – MIR extinction.** Panel (a) shows the synthetic MIR extinction curve of Hensley & Draine (2021) (in blue). The ISO observations of the Galactic center by Lutz et al. (1996) are overlaid in red. We also show the *Spitzer*, WISE and AKARI observations (Indebetouw et al., 2005; Xue et al., 2016). Panel (b) shows the absorbance of the most abundant ices, taken from the *Leiden Database for Ice*. Water ice data are from Öberg et al. (2007), CO, from Fraser & van Dishoeck (2004), and CO₂, from Bisschop et al. (2007). All ices have been measured at 15 K. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Silicate features.** The observed profiles of the Si–O stretching and O–Si–O bending silicate features, at 9.7 and 18 µm (cf. Sect. I.1.4), are not perfectly matching those of olivine and pyroxene. This is the reason of the introduction of the concept of *astrosilicates* by Draine & Lee (1984).

* Interstellar silicates are indeed a mixture of different varieties of compounds, with different stoichiometry, and probably metallic inclusions.

* Interstellar silicates are predominantly amorphous. The interstellar crystalline silicate fraction seems to be < 2% (Demyk et al., 1999; Kemper et al., 2004; Do-Duy et al., 2020).

There are uncertainties about the profile of the features and its potential variations between sightlines.

**The 9.7 µm feature** has on average a FWHM of ≈ 2.2 µm. A(V)/τ(9.7) ≈ 9 ± 1 toward the GC (Kemper et al., 2004), but is higher when averaged over sightlines: A(V)/τ(9.7) ≈ 19 (Roche & Aitken, 1984; Mathis, 1998). The synthetic extinction of Hensley & Draine (2021) has A(V)/τ(9.7) = 20.

**The 18 µm feature** is weaker than the 9.7 µm, making its characterization more uncertain. The depth ratio of the two features is τ(9.7)/τ(18) ≈ 2 (Chiar & Tielens, 2006).

**Ices.** In regions shielded from the stellar radiation, some molecules can freeze out to form icy mantles onto grains (cf. e.g. Boogert et al., 2015, for a review). The dominant species are H₂O, CO and CO₂. They are responsible for several MIR absorption bands, shown in Fig. II.12.b. These ice features are not observed in the diffuse ISM. They start appearing at higher values of A(V), different compositions having different melting points. They are observed in dense regions, toward molecular clouds, *Young Stellar Objects* (YSO) or AGNs.

### II.2.1.3 X-Rays

**X-ray halos.** Although the opacity of typical interstellar grains peaks in the UV (cf. e.g. Fig. I.19), grains extinct significantly X-rays. In this regime, photons have wavelengths approaching the size of

<figure><img src="image_198.png" /></figure>
the atoms in the grain. Dust grains, when present along the sightline of an X-ray point source (such as a low-mass X-ray binary), scatter the radiation at small angles, creating an *X-ray halo* (Overbeck, 1965; Smith & Dwek, 1998). The properties of this halo are complex, as they depend on the grain properties: composition, porosity and maximum size (e.g. Smith, 2008). Such studies are limited by the uncertainty on the distance of the intervening dust and the background source. They however confirm the low abundance of grains larger than ≈ 0.1 µm (e.g. Valencic & Smith, 2015).

<figure><img src="image_199.png" /><figcaption>FIGURE II.13 – *X-ray edges*. The two panels show the absorption and scattering dust opacities (blue and red), and the gas opacity (green). The opacities are expressed per unit ISM mass. The gas opacity includes only the elements of the grains they are compared to (i.e. C for graphite, and O, Si, Fe and Mg for silicates). The dust cross-sections are from Draine (2003c). We have assumed an MRN size distribution and a Galactic dustiness. The gas cross-section has been computed with the *python* interface of *X-ray DB*. We have assumed the Solar abundances of Asplund et al. (2009). Licensed under CC BY-SA 4.0.</figcaption></figure>

**X-ray absorption edges.** Atoms, whether in the gas phase, or locked-up in grains, exhibit X-ray absorption features at specific wavelengths, called *X-ray photoelectric edges* (cf. Fig. II.13). These edges correspond to the binding energies of the inner electrons, the letter (K or L in our case) corresponding to Bohr's orbitals (cf. Table I.2). The important point is that the energy and the spectral shape of these edges depend on the way the atom is paired (e.g. Draine, 2003c). It is thus possible, using X-ray spectroscopy, to differentiate atoms in the gas and dust phase, but also the crystalline structure of the grains (e.g. Lee et al., 2009a). For instance, Zeegers et al. (2017) studied the Si K edge along the line of sight of a Galactic X-ray binary. They were able to constrain the column density and the chemical composition of the silicate grains. This method was used to show that interstellar silicates are essentially Mg-rich, whereas the iron content is in metallic form (Costantini et al., 2012; Rogantini et al., 2019; Westphal et al., 2019). Finally, the crystalline fraction of silicates has been estimated to be in the range ≈ 11–15%, using X-ray spectra (Rogantini et al., 2019, 2020). This is significantly higher than the ≈ 2% upper limit derived from MIR spectroscopy (cf. Sect. II.2.1.2). This discrepancy might originate in the challenges of X-ray spectroscopy, which requires both high spectral resolution and high signal-to-noise ratios.
<figure><img src="image_200.png" /></figure>

II.2.1.4 Dichroic Extinction

The light from a background source seen through a cloud containing elongated grains, with their
rotation axis aligned along the magnetic field, is partially polarized (cf. Sect. I.2.2.5). In the MW, the
wavelength-dependent polarization fraction follows the empirical law of Serkowski (1973), shown
in Fig. II.14.a. It runs from the near-UV (NUV) to the NIR, peaking around $\lambda \approx 0.55$ µm. It is well
reproduced by models with elongated grains (cf. Fig. II.14.a and Guillet et al., 2018). The polarized
extinction fraction, $p(\lambda)$, is often quoted: $p(\lambda) \equiv C_{\text{pol}}(\lambda)/C_{\text{ext}}(\lambda)$, where $C_{\text{pol}}$ and $C_{\text{ext}}$ are the polarized
and total cross-sections.

The interstellar polarized extinction peaks around $\lambda_{\text{max}} \approx 0.55$ µm, and its fraction is $p(\lambda)/A(V) \lesssim 3\%/\text{mag}$ (Andersson et al., 2015).

<figure><img src="image_201.png" /></figure>

FIGURE II.14 – Polarized extinction and DIBs. Panel (a) shows the wavelength-dependent polar- ized extinction. The plotted quantity is the ratio between the polarized optical depth and the H col- umn density, $\sigma_{\text{pol}}(\lambda) \equiv \tau_{\text{pol}}(\lambda)/N_{\text{H}}$ The blue curve represents the synthetic, compromise fit of Whittet (2003). The red curve is the original Serkowski (1973) profile. Both have been normalized so that $p(V)/E(B-V) = 0.13$ (Hensley & Draine, 2021). The green curve shows the model E of Guillet et al. (2018). Panel (b) shows the average absorption spectrum of DIBs from the study of Jenniskens & Desert (1994). It is for given for a typical $E(B-V) = 1$. Licensed under CC BY-SA 4.0.

II.2.1.5 Diffuse Interstellar Bands

DIBs are ubiquitous absorption features in the ≈ 0.4–2 µm range (cf. Fig. II.14.b). They are too broad to originate in atoms or simple molecules. They have to come from large molecules and/or small grains. Over 500 of them have been detected in the ISM (Fan et al., 2019). They are empirically associated with dust, as their strength correlates with E(B–V) at low values, but they disappear in denser sightlines (e.g. Lan et al., 2015). To first order, DIBs correlate with each other, but there are some notable differences, suggesting that they have different carriers (Herbig, 1995). For instance, the so-called C<sub>2</sub> DIBs (Thorburn et al., 2003) appear to be found preferentially in diffuse molecular clouds. They remain largely unidentified, although four of them have been attributed to the ionized buckminsterfullerene, C<sub>60</sub>, a football-shaped carbon molecule (Campbell et al., 2015; Walker et al., 2015). The MIR transitions of this molecule, as well as C<sub>70</sub>, had been detected in the ISM, a few years before (Cami et al., 2010).

<figure><img src="image_202.png" /></figure>
<figure><img src="image_203.png" /></figure>

## II.2.2 Emission

As we have discussed in Sect. I.2.4, dust emits thermally in the IR. This thermal emission is also partially polarized. We will see in this section that there are also non-thermal emission components.

### II.2.2.1 Infrared Continuum and Features

**Observations of the diffuse ISM.** Fig. II.15 represents the NIR-to-cm SED of the diffuse Galactic ISM. Those are the observations used to constrain the dust models we will discuss in Sect. II.3. The challenge of building such a data set is ensuring that these fluxes correspond to the emission of the most diffuse regions of the MW, characterized by its H column density ($N_H \approx 10^{24} \text{ m}^{-2}$). The disk of the MW contains the densest regions (cf. Fig. II.4.b). It is also important to ensure avoiding denser regions, as grain properties evolve, probably due to the accretion of mantles (e.g. Ysard et al., 2015). These observations therefore focus at high Galactic latitude, *b*, and low $N_H$. For instance, Compiègne et al. (2011) used data at $|b| > 6^\circ$ and $N_H < 5.5 \times 10^{24} \text{ m}^{-2}$. Hensley & Draine (2021) gives a more a complete discussion about the homogenization of the different datasets. At these emission levels, there are several contaminations that need to be subtracted.

**The zodiacal foreground** is the MIR emission from the interplanetary dust in the Solar system disk.

**The CIB** that we have already discussed in Fig. I.28 is the accumulated emission of background galaxies. Its SED is very similar to the emission of the MW, and thus difficult to subtract.

**The CMB** is the mm emission shown in Fig. I.28.

<figure><img src="image_204.png" /></figure>

**FIGURE II.15 – Galactic diffuse ISM SED.** These observations are the typical constraints on the emission of dust models. The DIRBE data are from Dwek et al. (1997); the ISOCAM spectrum is from Flagey et al. (2006); the FIRAS spectrum has been reprocessed by Compiègne et al. (2011); and the Planck data are from Planck Collaboration et al. (2014b, 2015b). The polarized emission is from Planck Collaboration et al. (2015b): it is the linearly polarized intensity (Eq. I.54), $4\pi v P_\nu / N_H$. Licensed under CC BY-SA 4.0.

**MIR features.** The average MIR spectrum of the diffuse Galactic ISM is represented in Fig. II.15 (in blue). This particular spectrum corresponds to a smaller patch of the sky, and is scaled on the DIRBE
<figure><img src="image_205.png" /></figure>

12 µm photometry (Flagey et al., 2006; Compiègne et al., 2011). There are indeed no MIR spectroscopic all sky surveys. The MIR constraints of dust models prior to Compiègne et al. (2011) were only the DIRBE broadbands. This difference in MIR coverage has consequences on the derived abundances and profiles of the aromatic feature carriers that we will discuss on Sect. II.3. The profiles and relative intensities of the main aromatic features can alternatively be constrained by a combination of laboratory data and the emission of nearby gas-rich galaxies (e.g. Draine & Li, 2007; Hensley & Draine, 2021).

### II.2.2.2 Polarized Emission

We have seen in Sect. I.2.2.5 that elongated grains emit polarized IR radiation. Although the polarized submm emission of the ISM had been measured from various balloon-borne observatories (Benoît et al., 2004; Bennett et al., 2013), the Planck satellite provided the first all sky survey in several bands (Planck Collaboration et al., 2015b). These observations point toward one major result: large ISM grains have homogeneous properties. In other words, the IR emission can not originate in the mixing of several heterogeneous grain populations. Small grains have a negligible polarization effect. The models of Guillet et al. (2018), which account both for total intensity and polarization, indeed provide the best fit for a single population of large composite astrosilicates with a-C mantles. In parallel, Planck Collaboration et al. (2020b) showed that the polarized SED was consistent with a single MBB with $\beta \approx 1.5$ and $T \approx 20$ K.

The maximum polarization fraction at 850 µm is ≈ 20% (Planck Collaboration et al., 2020a).

### II.2.2.3 Non-Thermal Emission

**Spinning Grains.** The AME is a centimeter continuum excess that can not be accounted for by the extrapolation of dust models, free-free, synchrotron and molecular line emission (Fig. II.10). It was first detected in the MW (Kogut et al., 1996). Draine & Lazarian (1998a) promptly proposed that it was arising from the dipole emission of fastly rotating ultrasmall grains. The candidate carriers were thought to be PAHs. The Wilkinson Microwave Anisotropy Probe (WMAP; $\lambda \approx 3.2 - 13$ mm; 2001-2010) and Planck data of the Galaxy were successfully fitted with spinning dust models, including PAHs (Miville-Deschênes et al., 2008; Ysard & Verstraete, 2010; Planck Collaboration et al., 2011b). The cm SED in Fig. II.15 is dominated by spinning grain emission. In the MW, the AME correlates with all tracers of dust emission (e.g. Hensley et al., 2016). However, the AME intensity increases with the ISRF intensity, while PAHs are destroyed in high ISRFs. Hensley et al. (2016) thus proposed that the carriers of the AME could be nano-silicates, rather than PAHs. Refining the modeling of the MIR SED, Bell et al. (2019) showed that AME correlates better with the emission from charged PAHs, in the Galactic region $\lambda$-Orionis. This will be discussed in more details in Sect. III.2.2.2.

**Photoluminescence.** We have seen in Sect. II.1.2.4 that the ERE excess emission was thought to originate in the photoluminescence of dust grains. In reflection nebulae, ERE appears to be excited by FUV photons (11 eV $\lesssim$ hv $\lesssim$ 13.6 eV; e.g. Lai et al., 2017). It disappears if the exciting star has an effective temperature $T_{\text{eff}} \lesssim 10^4$ K. The conversion efficiency, that is the rate of photoluminescent photons per absorbed UV photon, seems to be around $\approx 1\%$. ERE being seen in reflection nebulae, it is expected to be a general property of interstellar grains. There is however a debate about the detection of ERE toward cirrus clouds and its conversion efficiency (cf. the discussion in Hensley & Draine, 2021). ERE is observed in C-rich PNe (containing predominantly carbonaceous grains) and not in O-rich PNe (containing predominantly silicates grains; Witt & Vrijh, 2004). The carriers should thus be carbon grains, such as PAHs.

<figure><img src="image_206.png" /></figure>
<figure><img src="image_207.png" /></figure>

## II.2.3 Elemental abundances in Grains

The logarithmic abundance of an element E, relative to H, is often noted:

$$ \epsilon(E) = 12 + \log\left(\frac{N_E}{N_H}\right), \qquad (II.8) $$

$N_E$ being its column density. The number abundance ratio can also be noted E/H instead of $N_E/N_H$, when it is not directly derived from the measure of a column density. An element in the ISM belongs either to the gas or to the dust phase. If we know the total or reference abundance of an element E in the ISM, we can thus infer its abundance locked in dust grains, by measuring its abundance in the gas phase. This difference is the depletion. The logarithmic depletion of an element E is defined as (Jenkins, 2009):

$$ \delta(E) = \epsilon(E_{\text{gas}}) - \epsilon(E_{\text{ref}}), \qquad (II.9) $$

The observable $\delta(E)$ is a measure of the ratio between the abundance of an element E observed in the gas phase to its total assumed abundance. The abundance of element E, locked in grains, is thus:

$$ \left(\frac{E_{\text{dust}}}{H}\right) = \left(\frac{E_{\text{ref}}}{H}\right) (1 - 10^{\delta(E)}). \qquad (II.10) $$

Note that, in Eq. (II.10), we do not differentiate the origin of H, as H is predominantly in the gas phase: $H_{\text{ref}} \approx H_{\text{gas}} \gg H_{\text{dust}}$.

### II.2.3.1 Measuring ISM abundances

**Solar abundances.** The abundance of elements and their isotopes are the most accurately known in the Solar system (e.g. Asplund et al., 2009, for a review). Those are thus used as a reference in the ISM. The abundances of the protosolar nebula, at the time the Sun formed, 4.56 Gyr ago, can be determined the two following ways.

**Meteorites**, analyzed with mass spectroscopy, provide the most precise abundances. The most primitive meteorites are the carbonaceous (CI) chondrites. The issue with meteorites is that the most volatile elements (i.e. the lightest ones and the noble gases) have been depleted due to high-temperature processes within the Solar nebula (e.g. Hellmann et al., 2020).

**Solar photosphere** absorption spectroscopy is less precise, as it requires some modeling. It however provides more reliable abundances of the volatile elements.

These abundances are compared in Fig. II.16. We see that both tracers are in very good agreement, except for the volatile elements. It is common to define the mass fractions of H, He, and elements heavier than He ($M_{ISM}$ being the total ISM mass):

$$ X \equiv \frac{M_H}{M_{ISM}}, \quad Y \equiv \frac{M_{He}}{M_{ISM}}, \quad Z \equiv \frac{M_{>He}}{M_{ISM}}, \quad \text{with } X+Y+Z=1. \qquad (II.11) $$

In the literature, the ratio Z is unanimously called *metallicity*. Some even call the elements heavier than He, *metals*, which is even worse, knowing what we have learned in Sect. I.1.3.1. This is one of the worst choices of terminology in the whole history of sciences. It is however difficult to avoid using the term *metallicity*. We will thus reluctantly use it in the rest of this manuscript.

**Present-day Solar abundances.** The abundances displayed in Fig. II.16 are present-day photospheric values. They are however not perfectly representative of the present-day abundances of the Solar neighborhood ISM. To go from the former to the latter, a factor +0.03 dex has to be added to the heavy element abundances of Fig. II.16 to account for diffusion in the Sun (Turcotte & Wimmer-Schweingruber, 2002). This provides protosolar abundances. Present-day abundances can then be inferred by modeling the chemical evolution of the MW during the last 4.56 Gyr (e.g. Chiappini et al.,

<figure><img src="image_208.png" /></figure>
<figure><img src="image_209.png" /></figure>

<figure><img src="image_210.png" /><figcaption>FIGURE II.16 – Solar abundances. The two lines show the elemental abundances relative to H of the Solar system, as a function of the atomic number, from Asplund et al. (2009). The red line and circles correspond to meteorites, and the blue line and circles correspond to the Solar photosphere. Licensed under CC BY-SA 4.0.</figcaption></figure>

2003; Bedell et al., 2018). This leads to correcting each element with a different factor, up to ≈ 0.2 dex (cf. e.g. Hensley & Draine, 2021, for the correction of the major dust constituents). The present-day Solar photospheric abundances are (Asplund et al., 2009):

$$X_{\odot} = 0.7381, \quad Y_{\odot} = 0.2485, \quad Z_{\odot} = 0.0134. \qquad (II.12)$$

To put it in words, three quarters of the gas mass is made of H, one quarter is made of He, and only 1.3% is made of heavy elements, in the MW. Besides H and He, the most abundant species in the ISM are O and C ($M_{O}/M_{ISM} \approx 8.0 \times 10^{-3}$ and $M_{C}/M_{ISM} \approx 2.8 \times 10^{-3}$). These abundances can be used as references in Eq. (II.9). Alternatively, B stars or young F and G stars can provide a more direct estimate of the abundances in nowadays ISM. These abundances are however more difficult to estimate accurately.

### II.2.3.2 Depletions

**The depletion strength.** The abundances in the gas phase are most reliably measured by absorption spectroscopy toward stars. Gas atoms in the neutral ISM are essentially in their ground state. Most of the corresponding transitions are in the UV ($\lambda = 0.0912-0.3 \mu m$). Jenkins (2009) compiled and homogenized the abundances of 17 elements measured along 243 sightlines, throughout the literature, to propose a unified representation of the depletions in the MW. Jenkins (2009) showed that the logarithmic depletions of each element are all linearly related, and controlled by a single parameter, $F_{\star}$, called the depletion strength factor:

$$\delta(E) \approx A_{E} \times F_{\star} + B_{E}. \qquad (II.13)$$

The factors $A_E$ and $B_E$ are empirically determined for each element. The depletion factor accounts for the fact that depletions are different along different sightlines. They however vary according to Eq. (II.13). This effect is due to dust growth in the ISM. It is supported by the good correlation between $F_{\star}$ and the average density of the ISM, demonstrated on Fig. II.17.a. When the density of the ISM
<figure><img src="image_211.png" /></figure>

increases, the collisional rate of a grain with heavy elements increases. A fraction of these elements stick on the grain surface and grow mantles.

* When $F_{\star} \approx 0$, we are in the most diffuse ISM, the depletion is $\delta(E) \approx B_{E}$.

* When $F_{\star} \approx 1$, such as toward $\zeta$ Oph, we are sampling the dense ISM. The amplitude of the depletion, $A_{E}$, reflects the composition of the grain mantles.

<figure><img src="image_212.png" /></figure>

FIGURE II.17 – Depletion variations within the MW. Panel (a): depletion factor (Eq. II.13) as a function of the average density, $\langle n_{\mathrm{H}} \rangle = (N_{\mathrm{H}1} + N_{\mathrm{H}2})/d$, where *d* is the distance to the star (Jenkins, 2009). The points represented in grey are those for which the uncertainty on $F_{\star}$ is greater than 0.07. Panel (b): depletion amplitude (from Eq. II.13) as a function of condensation temperature (Lodders, 2003; Jenkins, 2009). Licensed under CC BY-SA 4.0.

**Volatile and refractory elements.** Not all the most abundant elements in the ISM enter the dust composition. Some elements such as N or the noble gases are not significantly depleted. Fig. II.17.b shows a general relation between the depletion amplitude and the condensation temperature of the most abundant heavy elements. The most depleted elements are those which have a high condensation temperature. For that reason, elements are often classified in the two following categories.

**Volatile elements** are the elements with low condensation temperatures (C, N, O, noble gases). A moderate temperature is sufficient to remove them from the grains. These elements thus exist mainly in the gas phase.

**Refractory elements** are the elements with high condensation temperatures (those are essentially the metals). They can be present in grains up to high temperatures. Their abundance in the gas phase therefore exhibits large variations as a function of environment.

We note that, although C and O are two of the main dust constituents, these elements are classified as volatile. These elements are indeed mainly in the gas phase, as their depletion is moderate (cf. Fig. II.18.a). However, this modest depletion is sufficient to account for a large fraction of the dust mass.

**Inferred dust composition.** Since the individual depletion of each element can be inferred, it provides the unique prospect of constraining the average composition of dust grains. This composition changes with density, as mantles grow. Following Hensley & Draine (2021), we quote depletions for

<figure><img src="image_213.png" /></figure>
$F_{\star} = 0.5$ as they correspond to $\langle n_{H} \rangle \approx 0.3 \text{ cm}^{-3}$, which is appropriate for the diffuse ISM. From this vantage point, the dustiness of the diffuse Galactic ISM is:

$$Z_{\text{dust}} \equiv \frac{M_{\text{dust}}}{M_{\text{gas}}} \approx \frac{1}{126 \pm 20} \approx 0.0079 \pm 0.0012. \qquad (\text{II.14})$$

The dust-to-metal mass ratio is thus:

$$\text{DM} \equiv \frac{Z_{\text{dust}}}{Z} \approx \frac{1}{2 \pm 0.26} \approx 0.592 \pm 0.093. \qquad (\text{II.15})$$

The results of Jenkins (2009) indicate that the dustiness is $\approx 2.7$ times higher at $F_{\star} = 1$ than at $F_{\star} = 0$. The number and mass abundance in grains is represented in Fig. II.18. The carbonaceous-to-silicate mass ratio is:

$$\frac{M_{\text{C-dust}}}{M_{\text{Sil.}}} \approx 0.177 \pm 0.085. \qquad (\text{II.16})$$

Finally, we can have an idea of what the stoichiometry of silicates should be:

$$\text{SiO}_{6.6 \pm 2.5} \text{Mg}_{1.21 \pm 0.16} \text{Fe}_{1.13 \pm 0.14}. \qquad (\text{II.17})$$

We note it results in a higher Si:O ratio than in olivine (1:4) and pyroxene (1:3) (cf. Sect. I.1.4). It is currently difficult to understand where all the depleted oxygen is, even if it also forms various oxides, such as $Fe_2O_3$, $Al_2O_3$, etc.

<figure><img src="image_214.png" /><figcaption>FIGURE II.18 – MW dust composition inferred from depletions. Panel (a): number abundances of the main depleted elements, relative to H, in part per million (ppm), into dust (red) and gas (green). These values correspond to the MW, for $F_{\star} = 0.5$ (Table 2 of Hensley & Draine, 2021). The top of each histogram represent the total ISM abundance. Panel (b): mass fraction of the different elements locked in grains, in the MW, for $F_{\star} = 0.5$. Licensed under CC BY-SA 4.0.</figcaption></figure>

## II.2.4 Direct Measures

Direct characterization of interstellar grains is possible in a few particular situations: (i) presolar grain inclusions in meteorites; (ii) interstellar grains entering the heliosphere; or (iii) study of dust analogs in the laboratory.
### II.2.4.1 Meteorite Inclusions

**Grain identification.** Primitive meteorites contain presolar grains, that is grains that formed in the ISM before being incorporated in the early Solar nebula (e.g. Hoppe & Zinner, 2000). They are believed to have remained relatively unaltered since the formation of the Solar system. They can be identified by their isotopic anomalies (cf. Fig. II.19.a). Carbonaceous chondrites that we have mentioned in Sect. II.2.3.1 are of particular interest (e.g. Nittler et al., 2019). Interstellar grains identified in meteorites can have one of the following compositions (cf. Fig. II.19.b):

* Silicon carbide (SiC);

* Graphite (C sp²);

* Silicates (SiO₃⁻₄);

* Nanodiamonds (C sp³);

* Silicon nitride (Si₃N₄);

* Corundum (Al₂O₃);

* Spinel (MgAl₂O₄); and

* Titanium oxide (TiO₂).

The size of these grains ranges from a few tenths of nanometers to a few microns. Their isotopic ratios are consistent with condensation in the ejecta of SNe or Asymptotic Giant Branch (AGB) stars (cf. Fig. II.19.a).

<figure><img src="image_215.png" /></figure>

**FIGURE II.19 - Presolar grains in meteorites.** Panel (a) shows the oxygen isotopic ratios of meteoritic presolar grains, from the Presolar Grain Database of Washington University (Hynes & Gyngard, 2009; Stephan et al., 2020). Panel (b) shows pictures of presolar grains from primitive meteorites (Hoppe, 2010). The SiC grain is from a SN, the graphite from an AGB star or a SN, and the spinel and silicate grains are from AGB stars. Credit: (a) licensed under CC BY-SA 4.0; (b) courtesy of the Max Planck Institute for Chemistry, with permission from Peter HOPPE.

**Limitations.** Overall, the current analysis of presolar grains in meteorites suffers from several biases. The search for presolar grains in meteorites uses chemical treatments dissolving the silicate matrix (Draine, 2003a). It is the likely reason why: (i) most grains are crystalline stardust, (ii) why so few silicate grains are found, and (iii) why the smallest grains are not detected.
<figure><img src="image_216.png" /></figure>

II.2.4.2 Interplanetary Dust

**ISD flux and cometary dust.** We have seen in Sect. II.1.1.5 that several spacecrafts have collected *Interplanetary Dust Particles* (IDP) *in situ*. Among these IDPs, several grains have been shown to be of interstellar origins, because of the direction and speed of their flow. Cometary dust also provides important clues, as comets formed during the early epoch of the Solar system. They should contain pristine material. A class of IDPs called *Glass with Embedded Metals and Sulfides* (GEMS; Bradley, 1994; Keller & Messenger, 2008), are presolar. They have sizes ranging from ≈ 0.1 to 0.5 µm.

<figure><img src="image_217.png" /><figcaption>FIGURE II.20 – *Micrometeorite collection in Antarctica.* Collecting micrometeorites in the central Antarctic regions, at Dome C, in 2002. Credit: Jean Duprat, Cécile Engrand, courtesy of CNRS Photothèque.</figcaption></figure>

**Micrometeorites.** In addition to grain collection in space, IDPs entering the atmosphere become micrometeorites. These can be collected on Earth and analyzed in the laboratory. Antarctica is particularly interesting to that purpose, because of the absence of pollution and the possibility to sample the snow (cf. Fig. II.19; Rojas et al., 2021, for a review).

II.2.4.3 Laboratory Measurements

Dust analogs, that are solids we think are making up ISD, can be extensively studied in the labora-
tory (e.g. Henning, 2010, for a review). We can distinguish at least two general types of experiments:
(i) spectroscopic characterization; and (ii) reactivity and processing.

**Spectroscopic characterization.** Two general steps are required to perform such measures: (i) synthesizing the target compound; (ii) measuring its optical properties, usually in a rather narrow spectral regime. The details of these steps depend a lot on the nature of the compound, and on the spectral range explored. Fig. II.21 shows an example of a particular experimental device to measure PAH
<figure><img src="image_218.png" /><figcaption>**FIGURE II.21** – NASA Ames PAH experiment. A typical setup for a matrix-isolation experiment: (A) sample deposition configuration, (B) UV photolysis configuration, and (C) configuration for collecting the IR spectrum. Credit: Mattioda et al. (2020), with permission from Andy MATTIODA.</figcaption></figure>

properties in the IR, at NASA Ames. Different groups across the world specialize in such measures on PAHs (e.g. Useli-Bacchitta et al., 2010; Bauschlicher et al., 2018), carbon grains (e.g. Mennella et al., 1998; Dartois et al., 2016), silicates (e.g. Dorschner et al., 1995; Demyk et al., 2017a) and ices (e.g. White et al., 2009), among others. Fig. II.22 shows some of the results of the silicate study of Demyk et al. (2017a,b).

<figure><img src="image_219.png" /><figcaption>**FIGURE II.22** – Laboratory measurement of silicate opacities. These data are from the amorphous silicate samples of Demyk et al. (2017a). Among other parameters, this study samples the effects of: (a) temperature; and (b) iron fraction. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Grain reactivity and evolution.** Other experiments tackle the reactivity on grain surface (e.g. water formation; Dulieu et al., 2010). Grain evolution in the ISM is also studied. For instance, the photoproduction of a-C(:H) (Dartois et al., 2005), the ion absorption on carbon grains (Mennella et al., 2003), the processing under high energy (to mimic cosmic rays, e.g. Dartois et al., 2013). Some laboratory samples can even be exposed to space conditions (Kebukawa et al., 2019), onboard the *International Space Station* (ISS).
<figure><img src="image_220.png" /></figure>

## II.3 State-of-the-Art Dust Models

A dust model is defined by the abundance and size distribution of several grain components, characterized by their composition (PAH, graphite, silicates, etc.). We now review how the Galactic observables we have presented in Sect. II.2 are used to constrain modern dust models. These models are therefore specific to the Galactic diffuse ISM. When using such a model to interpret other observations, we can vary the intensity and spectral shape of the ISRF, to account for local variations. In principle, we can also vary the abundance of each component, and some parameters of the size distribution to fit observations of other systems. A dust model is a parametric framework that we can use to interpret any dust observable. There are however some limitations that we will discuss in Chap. III.

### II.3.1 Composition and Size Distributions of Different Models

There has been a large number of dust models in the past. We discuss here only some of the most recent ones (Zubko et al., 2004; Draine & Li, 2007; Compiègne et al., 2011; Siebenmorgen et al., 2014; Jones et al., 2017; Guillet et al., 2018).

#### II.3.1.1 Diversity in Composition

**Inherent degeneracies of dust models.** Different models make different choices in terms of composition. This is because, even with all the constraints we have listed in Sect. II.2, there are still numerous degeneracies. Several dust mixtures can fit the same observables. This has been best demonstrated by Zubko et al. (2004, hereafter ZDA04). ZDA04 fitted the UV-to-NIR extinction, IR emission and elemental depletions with different compositions, including: PAHs, graphite, different types of amorphous carbons, silicates, and composite grains. They also varied the reference abundances used to estimate elemental depletions. In the end, they showed 15 different dust mixtures providing satisfying fits to the Galactic diffuse ISM observables.

**Common compositional choices.** A dust model accounting for at least the UV-to-MIR extinction and the IR emission must have the following features.

* PAHs or small *a-C*(:H) are necessary to account for the aromatic features. Among the models we discuss here, only THEMIS also accounts for the 3.4 µm aliphatic feature. In addition, PAHs or small *a-C*(:H) account for a large fraction of the 2175 Å extinction bump.

* Silicate grains are necessary to account for the 9.7 and 18 µm silicate features. In addition, even if depletions are not actually fitted, they indicate that about 2/3 of dust mass must reside in some form of silicate grains (cf. Sect. II.2.3).

* Large carbon grains are necessary to account for the bulk of the FIR emission with a reasonable dustiness. Large, uncoated silicate grains are indeed not emissive enough to explain the FIR SED without requiring more heavy elements locked up in grains than what is available in the ISM. Graphite and, even more, *a-C*(:H) will increase the overall emissivity of the large grain mixture to the desirable level, using the second most abundant dust specie available.

In addition to these choices, grain mantles and/or inclusions can also enter the composition. Some have been explored by ZDA04. These are an essential part of the THEMIS model, which is designed as an evolution model (cf. Sect. II.3.2.2). The mantle thickness is indeed one of the parameters quantifying grain evolution through the ISM. Another important parameter is the shape of the grains. Elongated grains are necessary to account for the polarization in extinction and emission (cf. Sect. I.2.2.5). Siebenmorgen et al. (2014) designed a model accounting for the polarized extinction. The model of Guillet et al. (2018) is currently the only one also accounting for the polarized emission measured by Planck. Table II.2 summarizes the differences between the most recent dust models.

<figure><img src="image_221.png" /></figure>
<figure><img src="image_222.png" /></figure>

<table><caption>TABLE II.2 – *Comparison between different dust models.* The goal of this table is to illustrate the diversity of observational constraints and the possible choices of dust mixtures. In the first part of the table, some checkmarks are questionable. A given model may indeed not actually use a particular constraint, but end up being consistent with it, whereas another one may use it but provide an imperfect fit. In the second part, the difference between “small” and “large” grains is around a radius of $a \approx 10$ nm corresponding to the typical transition radius for stochastically heated grains (cf. Sect. I.2.4.3).</caption><thead><tr><th></th><th>Zubko et al. (2004, BARE-GR-S)</th><th>Draine & Li (2007)</th><th>Compiègne et al. (2011)</th><th>Siebenmorgen et al. (2014)</th><th>Jones et al. (2017)</th><th>Guillet et al. (2018)</th></tr></thead><tbody><tr><td colspan="7">Observational constraints accounted for</td></tr><tr><td>UV-to-NIR extinction</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Polarized extinction</td><td></td><td></td><td></td><td>✓</td><td></td><td>✓</td></tr><tr><td>MIR extinction</td><td></td><td></td><td>✓</td><td></td><td>✓</td><td>✓</td></tr><tr><td>Albedo</td><td></td><td></td><td>✓</td><td></td><td>✓</td><td>✓</td></tr><tr><td>NIR-to-mm emission</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Polarized emission</td><td></td><td></td><td></td><td></td><td></td><td>✓</td></tr><tr><td>Elemental depletions</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td colspan="7">Composition of the dust mixture</td></tr><tr><td>PAHs</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td></td><td></td></tr><tr><td>Small a-C(:H) grains</td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Large a-C(:H) grains</td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Small graphite grains</td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td></tr><tr><td>Large graphite grains</td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td></tr><tr><td>Small silicate grains</td><td>✓</td><td></td><td></td><td>✓</td><td></td><td></td></tr><tr><td>Large silicate grains</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Grain mantles</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td></tr><tr><td>Grain inclusions</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td></tr></tbody></table>

### II.3.1.2 Difference in Size Distributions

**Origin of the Size Distribution.** The size distribution of interstellar grains is a complex balance between the formation and destruction processes that we will discuss in Chap. IV. Two of these processes explain quite naturally two widely-used functional forms.

**Collisional fragmentation** of an initial distribution of large grains leads to a power-law size distribution with an index close to $\approx -3.5$, similar to the MRN size distribution (Eq. II.1; $f(a) \propto a^{-3.5}$). This result was demonstrated for asteroids by Hellyer (1970, with a different index, in his case). Dorschner (1982) explained the interstellar size distribution as a result of collisions in the circumstellar envelopes where grains are produced. A simplified demonstration of this process is given in Chap. 7 of Krügel (2003).

**Turbulent grain growth** results in a log-normal dust size distribution (Mattsson, 2020).

<figure><img src="image_223.png" /></figure>
<figure><img src="image_224.png" /></figure>

<figure><img src="image_225.png" /><figcaption>**FIGURE II.23** – *Size distribution of several dust models.* This figure shows the size distribution of the Zubko et al. (2004, BARE-GR-S), Compiègne et al. (2011) and THEMIS models. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Comparison Between Different Models.** Fig. II.23 compares the size distributions of three of the models we are discussing in this section. The displayed size distributions, $a \times f(a)$, are multiplied by $a^3$ so that they are mass-weighted. Although they manage to fit the same observables, these size distributions are quite different. ZDA04 adopt complex functional forms (Fig. II.23.a). Compiègne et al. (2011) use log-normal size distributions for PAHs and small a-C, and power-law with an exponential cut-off for large grains (Fig. II.23.b). It is the opposite for THEMIS, which uses log-normals for large grains and a power-law with an exponential cut-off for small a-C(:H) (Fig. II.23.c). Despite these differences, we notice the common features that we have listed in Sect. II.3.1.1.

* In the three panels of Fig. II.23, the aromatic feature carrying grains (in blue) are the smallest carbon grains. Their size distribution needs to peak around 3 Å $\lesssim a \lesssim 50$ Å, for these grains to fluctuate to high enough temperatures. Current observables are not accurate enough to distinguish differences in the shape of $f(a)$ in this range.

* Intermediate-size grains (5 nm $\lesssim a \lesssim 20$ nm) are necessary to account for the MIR continuum. ZDA04 use both graphite and silicate grains, resulting in the presence of the 9.7 and 18 µm features in emission. Compiègne et al. (2011) use a separate a-C log-normal component (in green). For the THEMIS model, the MIR continuum is accounted for by the tail of the small a-C(:H) (in blue).

* The large grain distributions (in blue and red) all peak around $a \approx 0.1$ µm. They need to drop sharply above this value, as larger grains tend to have lower equilibrium temperatures (cf. Sect. I.2.4.2) that would broaden the FIR emission peak of the SED.

## II.3.2 The Model Properties

Each model computes the panchromatic opacity, albedo and emissivity of its grain mixture. There are slight differences between different models, because they use different data sets and because the coverage of these data sets is not complete. The properties of the dust mixture of a model are simply the properties of its individual grains, integrated over the size distribution. For a given function $X(a)$,

<figure><img src="image_226.png" /></figure>
<figure><img src="image_227.png" /></figure>

we note:

$$ \langle X \rangle_a \equiv \int_{a_{-}}^{a_{+}} X(a) f(a) \, da. \qquad (II.18) $$

The general properties defined in Sect. I.2.2.3 and Sect. I.2.4.3 can therefore be generalized as:

$$ \langle m_{\text{dust}} \rangle_a = \left\langle \frac{4}{3} \pi a^3 \rho \right\rangle_a \qquad (II.19) $$

$$ \kappa_{\text{abs/sca}}(\lambda) = \frac{\langle \pi a^2 Q_{\text{abs/sca}}(a, \lambda) \rangle_a}{\langle m_{\text{dust}} \rangle_a} \qquad (II.20) $$

$$ g(\lambda) = \frac{\langle g(a, \lambda) \pi a^2 Q_{\text{sca}}(a, \lambda) \rangle_a}{\langle \pi a^2 Q_{\text{sca}}(a, \lambda) \rangle_a} \qquad (II.21) $$

$$ \epsilon_v(\lambda) = \frac{\left\langle 4\pi Q_{\text{abs}}(a, \lambda) \pi \int_0^\infty \frac{dP(T, a)}{dT} B_v(\lambda, T) dT \right\rangle_a}{\langle m_{\text{dust}} \rangle_a} \qquad (II.22) $$

### II.3.2.1 Extinction and Emission

**The opacity.** Fig. II.24.a compares the panchromatic opacity of different models. At first order, the four models are in good agreement.

* The discrepancies in the UV-to-NIR range are due to the different compositions. For instance, the THEMIS model tends to have more opaque material. The observations of the Galaxy being expressed per H atom, this model therefore requires a slightly lower dustiness.

* The Hensley & Draine (2021) model has a much flatter MIR continuum, to account for Galactic-center-type extinction curves (cf. Sect. II.2.1.2).

* At wavelengths longer than 100 µm, the THEMIS and Hensley & Draine (2021) models have a flatter opacity than the other ones. Planck constraints are indeed higher in this regime than the extrapolation of the COBE/FIRAS spectrum.

The bottom left panels of Fig. II.24 show the decomposition of the opacity of the THEMIS model.

**Scattering and absorption** are shown in Fig. II.24.b. We can see that scattering is dominant only in the NIR range. This scattering component mainly originates in large grains (cf. Sect. I.2.2.3).

**Carbonaceous and silicates** are shown in Fig. II.24.c. We can see that carbon grains dominate the UV and submm opacity. In the particular case we have displayed (the THEMIS model), silicate grains are coated with a-C(:H) mantles. The carbon component of a model made of bare grains would be sensibly higher.

**The SED.** Fig. II.24.d compares the SED of the same four models as previously. The shapes of these SEDs are relatively similar.

* The differences in the peak of the FIR SED is due to the difference in composition, mirroring the difference in opacity previously discussed.

* The difference in the level of the MIR continuum is due to the paucity of observational constraints in this regime.

* The difference in the level of the aromatic feature emission is due to: (i) the different dustiness; and (ii) the different sets of MIR constraints (cf. Sect. 3.1.1 of G21, for a discussion of this discrepancy).

The bottom right panels of Fig. II.24 show the decomposition of the THEMIS model in sizes and composition.

**Small and large grains** are shown in Fig. II.24.e. We can see that the small grains are responsible from the MIR emission, as they are stochastically heated (cf. Sect. I.2.4.3).

<figure><img src="image_228.png" /></figure>
<figure><img src="image_229.png" /></figure>

**Carbonaceous and silicates** are shown in Fig. II.24.f. Carbon grains are responsible for the entire MIR emission and for a small fraction of the FIR peak. Amorphous-carbon-coated-silicates are responsible for most of the FIR emission peak.

### II.3.2.2 The Fitted Constraints

We now demonstrate the fit of the observational constraints by one the models, THEMIS. We start by presenting this model in more depth.

**The THEMIS model.** It is a laboratory-data-based model. As we have previously discussed, it uses two populations of grains: (i) a-C(:H) grains with the optical properties of Jones (2012a,b,c); and (ii) a-Silicates with Fe and FeS inclusions and a-C(:H) mantles, whose optical properties have been computed by Köhler et al. (2015). The largest a-C(:H) are coated with a-C. A first version was presented by Jones et al. (2013) and updated by Jones et al. (2017). At the time this manuscript is being written, a new version is in preparation including the laboratory optical properties of silicates measured by Demyk et al. (2017a,b). It is an evolution model. The hydrogenation of a-C(:H), their size distribution, as well as the mantle thickness of the large grains are parameters evolving with the ISRF intensity and the density of the ISM.

**Discussion of the fit.** Fig. II.25 shows the fit of the diffuse Galactic ISM constraints by the THEMIS model.

**The extinction curve** is well fitted except in the 10 µm range (Fig. II.25.a). This region is the most problematic because of: (i) the uncertainty about the profiles of the astrophysical silicate mixture features, which is common to every model; and (ii) the uncertainty about the shape of the continuum in this range (cf. Sect. II.2.1.2). The synthetic observed extinction curve used to constrain this model is provided by Mathis (1990) without error bars.

**The elemental depletions** are relatively well fitted except for Fe (Fig. II.25.b). This is a common problem of contemporary dust models (cf. Sect. II.2.3).

**The albedo** is relatively well fitted (Fig. II.25.c). The problem is that the observational constraints themselves are rather scattered. Some constraints in the UV range are inconsistent. This is because albedo measurements come from a diversity of regions (DGL, reflection nebulae; cf. Sect. II.2.1.1), which are difficult to homogenize. In addition, the asymmetry parameter derived from these observations is rather high. The albedo is thus measured at the tail of the scattering phase function, which adds another layer of uncertainties.

**The emissivity** is well fitted (Fig. II.25.d). There are no problem with this component. This is important because this model is used to analyze the IR emission of galaxies and Galactic regions.

### II.3.3 Some Useful Quantities

We finish this chapter by listing a few quantities and formulae, useful to make simple estimates and approximations. Unless otherwise noted, these quantities are computed using the THEMIS dust model, and might slightly differ if another model is considered.

#### II.3.3.1 Grain Sizes, Areas and Masses

**For the MRN size distribution.** The grain surface is important for chemical reactions and for the photoelectric effect. For a MRN size distribution (Eq. II.1), the average grain surface is:

$$ \langle S_{\text{dust}} \rangle_a = \pi \int_{a_{-}}^{a_{+}} \underbrace{f(a)}_{\propto a^{-3.5}} a^2 da \propto \frac{1}{\sqrt{a_{+}}} - \frac{1}{\sqrt{a_{-}}} \approx \frac{1}{\sqrt{a_{-}}}. \qquad (II.23) $$

<figure><img src="image_230.png" /></figure>
<figure><img src="image_231.png" /><figcaption>FIGURE II.24 – Model opacity and emissivity. Panel (a) compares the total opacities of: the BARE-GRS mixture of Zubko et al. (2004); the Compiègne et al. (2011) model; the THEMIS model (Jones et al., 2017) and the synthetic observations of Hensley & Draine (2021). Panel (b) shows the decomposition of the opacity into absorption and scattering (for THEMIS). Panel (c) shows the decomposition of the opacity into silicates and carbonaceous grains (for THEMIS). Panel (d) compares the emissivity of the different dust models. Panel (e) shows the decomposition of the emissivity into large and small grains (for THEMIS). Panel (f) shows the decomposition of the emissivity into silicates and carbonaceous grains (for THEMIS). Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_232.png" /></figure>

<figure><img src="image_233.png" /><figcaption>FIGURE II.25 – THEMIS fit of the Galactic constraints. Panel (a): fit of the diffuse Galactic ISM extinction curve with the THEMIS model by Jones et al. (2013). The observations (black dots) are from Mathis (1990) and are not provided with uncertainties. Panel (b): albedo fit with the Jones et al. (2013) model. The observations are from: Lillie & Witt (1976); Morgan et al. (1976); Morgan (1980); Chlewicki & Laureijs (1988); Hurwitz et al. (1991); Witt et al. (1997). Panel (c): emissivity fit with the Jones et al. (2013) model. The observations are those of Fig. II.15. Licensed under CC BY-SA 4.0.</figcaption></figure>

where a is the grain radius, $f(a)$, the size distribution from Eq. (II.1), and $a_-$ and $a_+$, the minimum and maximum sizes ($a_+ \gg a_-$).

The grain surface is thus dominated by small grains.

The average grain volume is:

$$ \langle V_{\text{dust}} \rangle_a = \frac{4\pi}{3} \int_{a_{-}}^{a_{+}} \underbrace{f(a)}_{\propto a^{-3.5}} a^3 da \propto \sqrt{a_{+}} - \sqrt{a_{-}} \approx \sqrt{a_{+}}, \qquad (\text{II.24}) $$

For a given grain species, the volume is proportional to the mass, thus the average grain mass is $\langle m_{\text{dust}} \rangle_a \propto \sqrt{a_{+}}$, too.

<figure><img src="image_234.png" /></figure>
The grain mass is dominated by large grains.

**The case of the THEMIS model.** The size distribution of the THEMIS model (Fig. II.23.c) can be split into three components: (i) small a-C(:H); (ii) big a-C(:H); and (iii) silicates. Table II.3 gives the first moments of the size distribution of these three components as well as of the total. The first line indicates that most grains are small grains. This is also reflected in the last column of each line: the value of each parameter is very close to its value for small a-C(:H).

<table><caption>TABLE II.3 – Moments of the THEMIS size distribution. The number fractions are expressed in part per billion (ppb) and the masses in atomic mass unit (amu; Table B.2).</caption><thead><tr><th></th><th>Small a-C(:H)</th><th>Large a-C(:H)</th><th>a-Silicates</th><th>Total</th></tr></thead><tbody><tr><td>Grain number fraction, &lang;1&rang;<sub>a</sub></td><td>10<sup>9</sup> ppb</td><td>36 ppb</td><td>80 ppb</td><td>10<sup>9</sup> ppb</td></tr><tr><td>Average radius, &lang;a&rang;<sub>a</sub></td><td>0.54 nm</td><td>12 nm</td><td>13 nm</td><td>0.54 nm</td></tr><tr><td>Average area, &pi;&lang;a<sup>2</sup>&rang;<sub>a</sub></td><td>1.02 nm<sup>2</sup></td><td>1140 nm<sup>2</sup></td><td>1510 nm<sup>2</sup></td><td>1.03 nm<sup>2</sup></td></tr><tr><td>Average mass, 4/3&pi;&rho;&lang;a<sup>3</sup>&rang;<sub>a</sub></td><td>1040 amu</td><td>1.2 &times; 10<sup>8</sup> amu</td><td>2.5 &times; 10<sup>8</sup> amu</td><td>2890 amu</td></tr><tr><td>Mass fraction</td><td>23%</td><td>8.1%</td><td>69%</td><td>100%</td></tr></tbody></table>

**Mass fraction of small grains.** The mass fraction of aromatic feature emitting grains (i.e. a-C(:H) smaller than $a \approx 1.5$ nm; cf. Fig. 1 of G21) is $q_{AF} \approx 17\%$. Other models, using PAHs instead of small a-C(:H), use different values, because PAHs have more aromatic bonds per C atom than a-C(:H). A smaller PAH mass is thus required to account for the same aromatic band strength. The mass fraction of PAHs is $q_{PAH} \approx 4.6\%$ for the Zubko et al. (2004) and Draine & Li (2007) models, and $q_{PAH} \approx 7.7\%$ for the Compiègne et al. (2011) model. The difference between the two latter values is due to the different sets of MIR constraints they use (cf. Sect. II.2.2.1; see also Sect. 3.1.1 of G21, for a discussion). For the THEMIS model, the mass fraction of the grains responsible for the MIR continuum (i.e. small a-C(:H) with radii $a \gtrsim 1.5$ nm) is $q_{MIRcont} \approx 6\%$.

**Dustiness and other ratios.** Table II.4 gives various number and mass ratios for the THEMIS model. The dustiness and the dust-to-H mass ratios are equivalent quantities, there is just a factor $(1 - Y_{\odot} - Z_{\odot})$ difference. The third line tells us that there are about 2 dust grains per million H atoms in the ISM. The last line indicates that about 40% of the mass of heavy elements in the diffuse Galactic ISM is locked-up in dust grains.

<table><caption>TABLE II.4 – Dustiness and other ratios for the THEMIS model. For the second to fourth columns, the dust mass (M<sub>dust</sub>) or number (N<sub>dust</sub>) are those of the sole component. Therefore, the sum of the second to fourth columns is equal to the fifth column.</caption><thead><tr><th></th><th>Small a-C(:H)</th><th>Large a-C(:H)</th><th>a-Silicates</th><th>Total</th></tr></thead><tbody><tr><td>Dustiness, Z<sub>dust</sub> ≡ M<sub>dust</sub>/M<sub>gas</sub></td><td>1/800<br>≈ 1.3 × 10<sup>-3</sup></td><td>1/2260<br>≈ 4.4 × 10<sup>-4</sup></td><td>1/270<br>≈ 3.8 × 10<sup>-3</sup></td><td>1/183<br>≈ 5.5 × 10<sup>-3</sup></td></tr><tr><td>Dust-to-H mass ratio,<br>Y<sub>dust</sub> ≡ M<sub>dust</sub>/M<sub>H</sub></td><td>1/600<br>≈ 1.7 × 10<sup>-3</sup></td><td>1/1700<br>≈ 5.9 × 10<sup>-4</sup></td><td>1/200<br>≈ 5.0 × 10<sup>-3</sup></td><td>1/138<br>≈ 7.3 × 10<sup>-3</sup></td></tr><tr><td>Dust-to-H number ratio, N<sub>dust</sub>/N<sub>H</sub></td><td>5300 ppb</td><td>0.05 ppb</td><td>0.02 ppb</td><td>1905 ppb</td></tr><tr><td>Dust-to-metal mass ratio, Z<sub>dust</sub>/Z</td><td>1/11 ≈ 9%</td><td>1/30 ≈ 3%</td><td>28%</td><td>41%</td></tr></tbody></table>
<figure><img src="image_235.png" /></figure>

### II.3.3.2 Opacity and Emissivity

**Optical properties.** Table II.5 gives the opacity, $κ$, $τ/N_H$, and the albedo, $\tilde{\omega}$, at the central wavelengths of the photometric filters displayed in Fig. II.26. The opacity, $κ$ is expressed per mass of dust, whereas $τ/N_H$ is expressed per H atom in the gas phase. The two quantities are related by:

$$ \frac{\tau(\lambda)}{N_H} \equiv \kappa(\lambda) \times Z_{\text{dust}} \times \frac{m_H}{1 - Y_{\odot} - Z_{\odot}}, \qquad (\text{II.25}) $$

where $m_H$ is the mass of an H atom (Table B.2). Fig. II.26 displays a useful approximation, valid for $20 \ \mu\text{m} \lesssim \lambda \lesssim 1 \ \text{mm}$:

$$ \kappa(\lambda) \approx 0.64 \ \text{m}^2/\text{kg} \times \left( \frac{250 \ \mu\text{m}}{\lambda} \right)^{1.79}. \qquad (\text{II.26}) $$

<figure><img src="image_236.png" /><figcaption>FIGURE II.26 – IR approximation of the opacity. This figure shows the extinction opacity of the THEMIS model. We have highlighted the different photometric bands we quote in Table II.5. We also show the power-law approximation of the IR opacity (in magenta; Eq. II.26). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Heating regimes.** It is important to understand which regime is dominated by large grains at equilibrium with the ISRF (Sect. I.2.4.2), and which one is dominated by small, stochastically heated grains (Sect. I.2.4.3). Fig. II.27.a shows the variation of the SED as a function of the ISRF intensity, U (Sect. I.2.4.2). We can see that when the intensity increases, the emission by large grains shifts to shorter wavelengths, as their equilibrium temperature increases. On the contrary, the emission by small, out-of-equilibrium grains stays constant, as these grains are heated by single photon events. Only their total intensity increases, which is hidden in Fig. II.27.a by the normalization of the intensity. We can estimate the transition wavelength, $\lambda_{\text{trans}}(\text{U})$, as the wavelength where the intensity of the small and large grains are equal. This is demonstrated in Fig. II.27.b. The values of $\lambda_{\text{trans}}(\text{U})$ for the grid of U displayed in Fig. II.27.a is given in Table II.6.

**Emissivity.** Table II.7 gives the emissivity of the THEMIS model. The emissivity is proportional to U. We give only the value for $U = 1$. We quote the following two values.

<figure><img src="image_237.png" /></figure>
<figure><img src="image_238.png" /></figure>

<table><caption>TABLE II.5 – *Optical properties of the THEMIS model.* This table gives several optical properties at each of the center of the photometric bands displayed in Fig. II.26. The opacity, $κ$, is the cross-section per mass of the dust component, whereas $τ/N_H$ is the cross-section per H atom in the gas phase. The albedo is $\tilde{\omega} = κ_{sca}/κ$ (Eq. I.40), and g is the asymmetry parameter (Eq. I.43).</caption><thead><tr><th></th><th></th><th>Small a-C(:H)</th><th>Large a-C(:H)</th><th>a-Silicates</th><th>Total</th></tr></thead><tbody><tr><td rowspan="4">U band<br>(0.36 µm)</td><td>κ</td><td>5934 m<sup>2</sup>/kg</td><td>9980 m<sup>2</sup>/kg</td><td>6107 m<sup>2</sup>/kg</td><td>6381 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>1.7 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>9.8 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>5.1 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>7.8 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>3.4 × 10<sup>-3</sup></td><td>0.43</td><td>0.60</td><td>0.45</td></tr><tr><td>g</td><td>0.19</td><td>0.57</td><td>0.56</td><td>0.56</td></tr><tr><td rowspan="4">B band<br>(0.44 µm)</td><td>κ</td><td>3245 m<sup>2</sup>/kg</td><td>8549 m<sup>2</sup>/kg</td><td>5180 m<sup>2</sup>/kg</td><td>5008 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>9.1 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>8.4 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>4.3 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>6.1 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>3.4 × 10<sup>-3</sup></td><td>0.46</td><td>0.62</td><td>0.51</td></tr><tr><td>g</td><td>0.18</td><td>0.55</td><td>0.54</td><td>0.54</td></tr><tr><td rowspan="4">V band<br>(0.55 µm)</td><td>κ</td><td>2023 m<sup>2</sup>/kg</td><td>7027 m<sup>2</sup>/kg</td><td>4272 m<sup>2</sup>/kg</td><td>3979 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>5.7 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>6.9 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>3.6 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>4.8 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>2.8 × 10<sup>-3</sup></td><td>0.48</td><td>0.64</td><td>0.54</td></tr><tr><td>g</td><td>0.16</td><td>0.54</td><td>0.53</td><td>0.53</td></tr><tr><td rowspan="4">R band<br>(0.66 µm)</td><td>κ</td><td>1395 m<sup>2</sup>/kg</td><td>5789 m<sup>2</sup>/kg</td><td>3543 m<sup>2</sup>/kg</td><td>3231 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>3.9 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>5.7 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>3.0 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>3.9 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>2.3 × 10<sup>-3</sup></td><td>0.48</td><td>0.65</td><td>0.56</td></tr><tr><td>g</td><td>0.14</td><td>0.53</td><td>0.51</td><td>0.52</td></tr><tr><td rowspan="4">I band<br>(0.80 µm)</td><td>κ</td><td>920 m<sup>2</sup>/kg</td><td>4487 m<sup>2</sup>/kg</td><td>2762 m<sup>2</sup>/kg</td><td>2479 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>2.6 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>4.4 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>2.3 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>3.0 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>1.7 × 10<sup>-3</sup></td><td>0.48</td><td>0.66</td><td>0.58</td></tr><tr><td>g</td><td>0.12</td><td>0.52</td><td>0.50</td><td>0.50</td></tr><tr><td rowspan="4">J band<br>(1.25 µm)</td><td>κ</td><td>398 m<sup>2</sup>/kg</td><td>2380 m<sup>2</sup>/kg</td><td>1453 m<sup>2</sup>/kg</td><td>1286 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>1.1 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>2.4 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>1.2 × 10<sup>-26</sup> m<sup>2</sup>/H</td><td>1.6 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>7.5 × 10<sup>-4</sup></td><td>0.45</td><td>0.70</td><td>0.61</td></tr><tr><td>g</td><td>0.07</td><td>0.49</td><td>0.46</td><td>0.47</td></tr><tr><td rowspan="4">H band<br>(1.60 µm)</td><td>κ</td><td>260 m<sup>2</sup>/kg</td><td>1636 m<sup>2</sup>/kg</td><td>982 m<sup>2</sup>/kg</td><td>869 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>7.3 × 10<sup>-28</sup> m<sup>2</sup>/H</td><td>1.6 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>8.2 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>1.1 × 10<sup>-26</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>4.6 × 10<sup>-4</sup></td><td>0.42</td><td>0.71</td><td>0.61</td></tr><tr><td>g</td><td>0.05</td><td>0.48</td><td>0.44</td><td>0.45</td></tr><tr><td rowspan="4">K band<br>(2.18 µm)</td><td>κ</td><td>156 m<sup>2</sup>/kg</td><td>1005 m<sup>2</sup>/kg</td><td>573 m<sup>2</sup>/kg</td><td>512 m<sup>2</sup>/kg</td></tr><tr><td>τ/N<sub>H</sub></td><td>4.4 × 10<sup>-28</sup> m<sup>2</sup>/H</td><td>9.9 × 10<sup>-28</sup> m<sup>2</sup>/H</td><td>4.8 × 10<sup>-27</sup> m<sup>2</sup>/H</td><td>6.2 × 10<sup>-27</sup> m<sup>2</sup>/H</td></tr><tr><td>ω</td><td>2.5 × 10<sup>-4</sup></td><td>0.37</td><td>0.68</td><td>0.58</td></tr><tr><td>g</td><td>0.03</td><td>0.45</td><td>0.42</td><td>0.43</td></tr></tbody></table>

<table><caption>TABLE II.6 – Transition wavelengths between small and large grain emission.</caption><thead><tr><th></th><th>U=0.1</th><th>U=1</th><th>U=10</th><th>U=100</th><th>U=1000</th><th>U=10<sup>4</sup></th></tr></thead><tbody><tr><td>λ<sub>trans</sub>(U)</td><td>88 µm</td><td>62 µm</td><td>43 µm</td><td>31 µm</td><td>22 µm</td><td>17 µm</td></tr></tbody></table>

<figure><img src="image_239.png" /></figure>
<figure><img src="image_240.png" /></figure>

<figure><img src="image_241.png" /><figcaption>FIGURE II.27 – Effect of U on the SED. Panel (a) shows the SED of the THEMIS model for different ISRF intensities, U. Panel (b) shows the intensity of the THEMIS model for U = 1 (Sect. I.2.4.2). We have decomposed the emission into large (a-Silicates and big a-C(:H)) and small (small a-C(:H)) grains. We show the transition wavelength between the two heating regimes, $\lambda_{trans}$ (U). In both panels, the y-axis is the intensity normalized by its bolometric integral. Each displayed SED therefore emits the same total intensity. Licensed under CC BY-SA 4.0.</figcaption></figure>

The emissivity, per se, integrated over the wavelengths, is the emitted power per unit dust mass:

$$ \epsilon \equiv \int_{0}^{\infty} \epsilon_{v}(v) \, dv. \qquad (II.27) $$

The emitted power per H atom, also integrated over the wavelengths, is expressed per H atom in the gas phase:

$$ \frac{4\pi I}{N_H} = \int_{0}^{\infty} \frac{4\pi I_v(v)}{N_H} \, dv. \qquad (II.28) $$

These two quantities are related by:

$$ \frac{4\pi I}{N_H} = \epsilon \times Z_{\text{dust}} \times \frac{m_H}{1 - Y_{\odot} - Z_{\odot}}. \qquad (II.29) $$

<table><caption>TABLE II.7 – Emissivity of the THEMIS model. These emissivities are proportional to U.</caption><thead><tr><th></th><th>Small a-C(:H)</th><th>Large a-C(:H)</th><th>a-Silicates</th><th>Total</th></tr></thead><tbody><tr><th>ε</th><td>98 × U L<sub>☉</sub> / M<sub>☉</sub></td><td>25 × U L<sub>☉</sub> / M<sub>☉</sub></td><td>97 × U L<sub>☉</sub> / M<sub>☉</sub></td><td>221 × U L<sub>☉</sub> / M<sub>☉</sub></td></tr><tr><th>4πI/N<sub>H</sub></th><td>2.3 × 10<sup>-31</sup> × U W/H</td><td>6.0 × 10<sup>-32</sup> × U W/H</td><td>2.3 × 10<sup>-31</sup> × U W/H</td><td>5.2 × 10<sup>-31</sup> × U W/H</td></tr></tbody></table>

<figure><img src="image_242.png" /></figure>
<figure><img src="image_243.png" /></figure>
<figure><img src="image_244.png" /></figure>

# Chapter III

# The Grain Properties of Nearby Galaxies

*Without data, you're just another person with an opinion.*

(Attributed to W. Edwards DEMING)

## Contents

<table><tr><td>III.1 Spectral Energy Distribution Modeling</td><td>100</td></tr><tr><td>III.1.1 Radiative Transfer</td><td>100</td></tr><tr><td>III.1.1.1 Definition of the Main Radiative Transfer Quantities</td><td>100</td></tr><tr><td>The moments of the specific intensity.</td><td>100</td></tr><tr><td>Specific energy density.</td><td>101</td></tr><tr><td>Emission coefficient and emissivity.</td><td>101</td></tr><tr><td>Extinction coefficient and opacity.</td><td>102</td></tr><tr><td>Mean free path.</td><td>102</td></tr><tr><td>III.1.1.2 The Radiative Transfer Equation</td><td>103</td></tr><tr><td>Propagation in vacuum.</td><td>104</td></tr><tr><td>Emission only.</td><td>104</td></tr><tr><td>Absorption only.</td><td>104</td></tr><tr><td>Emission and absorption.</td><td>105</td></tr><tr><td>Scattering and absorption with central illumination.</td><td>105</td></tr><tr><td>Scattering and absorption with uniform illumination.</td><td>106</td></tr><tr><td>Scattering and absorption with external illumination.</td><td>108</td></tr><tr><td>III.1.1.3 Approximations for Clumpy Media</td><td>108</td></tr><tr><td>Effective optical depth of a clumpy medium.</td><td>108</td></tr><tr><td>The mega-grains approximation.</td><td>109</td></tr><tr><td>Escape fractions for a clumpy medium.</td><td>109</td></tr><tr><td>III.1.1.4 Rigorous Solutions</td><td>109</td></tr><tr><td>Setting the model.</td><td>109</td></tr><tr><td>The principle of Monte-Carlo radiative transfer.</td><td>111</td></tr><tr><td>Numerical method to randomly draw photons.</td><td>111</td></tr><tr><td>Example of a clumpy medium.</td><td>112</td></tr><tr><td>III.1.2 Approximate Treatments of the Mixing of Physical Conditions</td><td>116</td></tr><tr><td>III.1.2.1 The Historical Model: the MBB</td><td>116</td></tr><tr><td>III.1.2.2 A Phenomenological, Composite Approach</td><td>117</td></tr><tr><td>The average starlight intensity.</td><td>118</td></tr><tr><td>Constraining the dust properties.</td><td>119</td></tr><tr><td>Limitations of the composite approach.</td><td>120</td></tr><tr><td>An alternative distribution.</td><td>120</td></tr></table>

<figure><img src="image_245.png" /></figure>
<figure><img src="image_246.png" /></figure>

<table>
  <tbody>
    <tr>
      <td>III.1.2.3 Panchromatic Empirical SED Models</td>
      <td>122</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Stellar SEDs.</td>
      <td>122</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Putting everything together.</td>
      <td>124</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Limitations of empirical panchromatic models.</td>
      <td>126</td>
    </tr>
    <tr>
      <td>III.1.2.4 The Matryoshka Effect</td>
      <td>128</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Demonstration on the LMC.</td>
      <td>128</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Generalization.</td>
      <td>128</td>
    </tr>
    <tr>
      <td>III.1.3 Application to Nearby Galaxies</td>
      <td>128</td>
    </tr>
    <tr>
      <td>III.1.3.1 The Different Types of Galaxies</td>
      <td>128</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The outdated galaxy morphological classification.</td>
      <td>128</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Galactic properties that matter to ISM studies.</td>
      <td>130</td>
    </tr>
    <tr>
      <td>III.1.3.2 Large-Scale Dust Distribution in Galaxies</td>
      <td>131</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Homogeneous multi-wavelength data sets.</td>
      <td>131</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Properties of individual galaxies.</td>
      <td>134</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Identifying dust heating sources.</td>
      <td>134</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Large-scale radiative transfer models of galaxies.</td>
      <td>135</td>
    </tr>
    <tr>
      <td>III.1.3.3 Constraining the Grain Opacity</td>
      <td>135</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Studies of the emissivity index.</td>
      <td>135</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Grain opacity in the LMC.</td>
      <td>136</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Confirmation in other systems.</td>
      <td>137</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The opacity in nearby galaxies.</td>
      <td>138</td>
    </tr>
    <tr>
      <td>III.1.3.4 Constraining the Size Distribution</td>
      <td>138</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Grain processing by shock waves.</td>
      <td>138</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The modeling strategy.</td>
      <td>138</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The grain size distribution in four dwarf galaxies.</td>
      <td>139</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Consequence on the extinction curves.</td>
      <td>140</td>
    </tr>
    <tr>
      <td>III.2 Studies Focussing on Specific Spectral Domains</td>
      <td>141</td>
    </tr>
    <tr>
      <td>III.2.1 Scrutinizing Mid-IR Spectra</td>
      <td>141</td>
    </tr>
    <tr>
      <td>III.2.1.1 The Aromatic Feature Spectrum</td>
      <td>142</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">MIR spectra of galaxies.</td>
      <td>142</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Laboratory and theoretical PAH physics.</td>
      <td>142</td>
    </tr>
    <tr>
      <td>III.2.1.2 Spectral Decomposition Methods</td>
      <td>144</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Calibrating feature properties.</td>
      <td>145</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Fitting every feature at once.</td>
      <td>146</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Alternative methods.</td>
      <td>146</td>
    </tr>
    <tr>
      <td>III.2.1.3 PAH Band Ratio Studies</td>
      <td>151</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Observed band ratios in galaxies.</td>
      <td>152</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Effects of ionization and size.</td>
      <td>152</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The particular case of low-metallicity environments.</td>
      <td>154</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">UIBs as diagnostics of the physical conditions.</td>
      <td>155</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Other properties.</td>
      <td>156</td>
    </tr>
    <tr>
      <td>III.2.1.4 Variations of the Aromatic Feature Strength</td>
      <td>157</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Effect of ISRF hardness.</td>
      <td>158</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Effect of metallicity.</td>
      <td>158</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">The absence of metallicity threshold.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>III.2.2 Long-Wavelength Properties</td>
      <td>159</td>
    </tr>
    <tr>
      <td>III.2.2.1 The Elusive Submillimeter Excess</td>
      <td>160</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Possible explanations.</td>
      <td>160</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Empirical properties of the excess.</td>
      <td>161</td>
    </tr>
    <tr>
      <td style="padding-left: 2em;">Reality of the phenomenon.</td>
      <td>161</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_247.png" /></figure>
<figure><img src="image_248.png" /></figure>

<table><tbody><tr><td>III.2.2.2 The Anomalous Microwave Emission</td><td>162</td></tr><tr><td>The AME in extragalactic environments.</td><td>163</td></tr><tr><td>Controversy about the carriers of the AME.</td><td>163</td></tr><tr><td>Correlation with charged PAHs.</td><td>163</td></tr><tr><td>III.3 Dust in Relation with the Gaseous and Stellar Contents</td><td>165</td></tr><tr><td>III.3.1 The Phases of the ISM</td><td>165</td></tr><tr><td>The cooling function.</td><td>165</td></tr><tr><td>III.3.1.1 The Neutral Atomic Gas</td><td>166</td></tr><tr><td>The photoelectric heating.</td><td>166</td></tr><tr><td>The two stable neutral atomic phases.</td><td>166</td></tr><tr><td>III.3.1.2 The Ionized Gas</td><td>168</td></tr><tr><td>The Hot Ionized Medium (HIM).</td><td>168</td></tr><tr><td>H II regions.</td><td>168</td></tr><tr><td>The Warm Ionized Medium (WIM).</td><td>170</td></tr><tr><td>III.3.1.3 The Molecular Gas</td><td>170</td></tr><tr><td>Molecular hydrogen formation.</td><td>170</td></tr><tr><td>The diffuse molecular gas.</td><td>170</td></tr><tr><td>Photodissociation regions.</td><td>170</td></tr><tr><td>Dense molecular clouds.</td><td>171</td></tr><tr><td>III.3.2 Dust as a Diagnostic Tool</td><td>171</td></tr><tr><td>III.3.2.1 Dust to Study Star Formation</td><td>171</td></tr><tr><td>Star formation rates.</td><td>171</td></tr><tr><td>Resolving star formation.</td><td>173</td></tr><tr><td>III.3.2.2 Photodissociation Regions</td><td>173</td></tr><tr><td>PDR Properties.</td><td>173</td></tr><tr><td>Photoelectric heating.</td><td>173</td></tr><tr><td>III.3.2.3 The Molecular Gas and its Dark Layer</td><td>174</td></tr><tr><td>The dark gas.</td><td>174</td></tr><tr><td>Pressure and radiation field.</td><td>175</td></tr></tbody></table>

In this chapter, we review how the models presented in Sect. II.3 are used to derive the grain properties of nearby galaxies. The term “dust properties” is vague. In the literature, it often indistinctively encompasses the three following categories (Galliano, Galametz, & Jones, 2018).

**The dust mixture constitution** is characterized by:

* the chemical composition of the bulk material and its stoichiometry (Sect. I.1.4);

* the structure of the grains (crystalline, amorphous, porous, aggregated, etc.; Sect. I.2.2.4);

* the presence of heterogeneous inclusions (Sect. I.2.2.4);

* the presence of organic and/or icy mantles (Sect. I.2.2.4);

* the shape of the grains (Sect. I.2.2.5);

* their size distribution (Sect. II.3.2);

* their abundance relative to the gas (Sect. II.2.3).

**The dust physical conditions** are the state a grain enters, when exposed to a particular environment:

* thermal excitation due to (i) radiative heating (equilibrium or stochastic; Sects. I.2.4.2 – I.2.4.3); and/or (ii) collisional heating in a hot plasma (Sect. I.2.4.4);

* grain charging by exchange of electrons with the gas (Sect. III.3.1.1);

* alignment of elongated grains with the magnetic field (Sect. I.2.2.5);

* grain rotation (Sect. I.2.2.5; Sect. II.2.2.3).

<figure><img src="image_249.png" /></figure>
<figure><img src="image_250.png" /></figure>

**The dust observables** arise from a grain mixture experiencing a particular set of physical conditions.

**Emission** of partially polarized components (Sect. II.2.2.2): (i) a thermal continuum (IR to mm; Sect. II.2.2.1); (ii) molecular and solid-state features (MIR; Sect. II.2.2.1); (iii) a possible microwave emission (cm; Sect. II.2.2.3); (iv) a possible luminescence (visible; Sect. II.2.2.3).

**Absorption** of the light from a background source by (Sect. II.2.1): (i) a continuum (X-ray to MIR; Sect. II.2.1.1); (ii) atomic, molecular and solid-state features (X-rays, UV and MIR; Sect. II.2.1.2), including DIBs (visible; Sect. II.2.1.5) and ices (MIR; Sect. II.2.1.2). The escaping light can be partially polarized as a result (Sect. II.2.1.4).

**Scattering** of the light from a bright source in our direction, and its polarization (X-rays to NIR; Sect. II.2.1).

**Depletion** patterns seen through the gas-phase elemental abundances (Sect. II.2.3).

# III.1 Spectral Energy Distribution Modeling

SED modeling is one of the main methods to empirically derive the dust properties of a region or a galaxy (cf. e.g. Galliano et al., 2018, for a review). The inherent complexity of astrophysical sources requires to account for the diversity of physical conditions within the studied region. The treatment of radiative transfer, even in an extremely approximated fashion, is thus necessary.

## III.1.1 Radiative Transfer

Radiative transfer is the method solving the propagation of multiple rays of light, emitted by one or several sources, through a macroscopic heterogeneous medium. It accounts for the scattering, absorption and emission, at each point and along each direction, in the studied region.

### III.1.1.1 Definition of the Main Radiative Transfer Quantities

Radiative transfer deals with several quantities that are often improperly defined or mixed together in the literature: intensity, flux, emissivity, brightness, etc. We have already seen some of these quantities in Sect. I.2.4. We now define them and explicit their differences (cf. Chap. 1 of Rybicky & Lightman, 1979, for a complete review). In what follows, we assume stationary systems. The time variable, $t$, is used only to denote constant rates.

**The moments of the specific intensity.** The primary radiative transfer quantity is the *specific intensity* or *brightness* (cf. Fig. III.1.a):

$$I_v(v, \vec{r}, \theta, \phi) = \frac{dE}{dt dAd\Omega dv} \qquad (III.1)$$

The specific intensity is the electromagnetic energy, E, per unit time, $t$, area, A, solid angle, $\Omega$, and frequency, v. It therefore quantifies the infinitesimal power carried by a monochromatic light ray. This quantity depends on the position in the region, $\vec{r}$, and on the direction of propagation, $(\theta, \phi)$. We adopt the spherical coordinate conventions used in physics (Fig. III.1; cf. Appendix C.1):

* the polar angle, $0 \le \theta < \pi$, is the angle with the z axis;

* the azimuthal angle, $0 \le \phi < 2\pi$, is the rotation angle in the (x, y) plane;

* the solid angle element is $d\Omega \equiv d\cos\theta d\phi = \sin\theta d\theta d\phi$, with:

$$\iint_{\text{sphere}} d\Omega = \int_{0}^{2\pi} d\phi \int_{-1}^{1} d\cos\theta = 4\pi. \qquad (III.2)$$

The first moments of the specific intensity, relative to its angular distribution, are physically meaningful.

<figure><img src="image_251.png" /></figure>
<figure><img src="image_252.png" /></figure>

**The mean intensity** is the zeroth order moment of Eq. (III.1):

$$J_v(v, \vec{r}) = \frac{1}{4\pi} \iint_{\text{sphere}} I_v(v, \vec{r}, \theta, \phi) d\Omega. \qquad (III.3)$$

It is the specific intensity averaged over all directions. It is often used to quantify the ISRF, $4\pi J_v$ accounting for rays coming from all directions. In the isotropic case, we have $I_v(v, \vec{r}, \theta, \phi) = J_v(v, \vec{r}), \forall (\theta, \phi)$.

**The net flux** (cf. Fig. III.1.b) is the first order moment of Eq. (III.1):

$$F_v(v, \vec{r}) = \iint_{\text{sphere}} I_v(v, \vec{r}, \theta, \phi) \cos \theta d\Omega. \qquad (III.4)$$

It represents the monochromatic power per unit area passing through a surface element perpendicular to $\hat{z}$. The $\cos\theta$ factor is there to account for the reduction of the density of rays that are not perpendicular to the surface. In the isotropic case, $F_v = J_v(2\pi \int_{-1}^{1} \cos\theta d\cos\theta = 0)$, because there is the same amount of flux passing through the area in both directions.

**The radiation pressure** (cf. Fig. III.1.c) is the second order moment of Eq. (III.1):

$$p_v(v, \vec{r}) = \frac{1}{c} \iint_{\text{sphere}} I_v(v, \vec{r}, \theta, \phi) \cos^2 \theta d\Omega \qquad (III.5)$$

It is the *momentum flux* carried by the photons, as $p = h\nu/c$ is the momentum of a single photon. The $\cos^2\theta$ term comes from two sources: (i) one $\cos\theta$ comes from the reduced fraction of inclined rays, similar to the flux; (ii) the second $\cos\theta$ factor comes from the fact the pressure is the momentum vector component that is perpendicular to the surface.

**Specific energy density.** The radiative energy within a volume element at a given time is the specific energy density:

$$u_v(v, \vec{r}, \theta, \phi) = \frac{dE}{dV d\Omega dV} = \frac{dE}{cdtdAd\Omega dV}. \qquad (III.6)$$

The second equality comes from the fact that $dV = ctdA$, $dV$ being a volume element (cf. Fig. III.1.d). The specific intensity is a power per unit area, whereas the energy density is an energy per unit volume. Both quantities are linked, combining Eq. (III.1) and Eq. (III.6):

$$u_v(v, \vec{r}, \theta, \phi) = \frac{I_v(v, \vec{r}, \theta, \phi)}{c}. \qquad (III.7)$$

If we integrate Eq. (III.6) over all directions, we get, using Eq. (III.3):

$$U_v(v, \vec{r}) = \iint_{\text{sphere}} u_v(v, \vec{r}, \theta, \phi) d\Omega = \frac{4\pi}{c} J_v(v, \vec{r}). \qquad (III.8)$$

**Emission coefficient and emissivity.** The monochromatic emission coefficient is the power radiated in a given direction, per unit volume and frequency:

$$j_v(v, \vec{r}, \theta, \phi) = \frac{dE_{\text{em}}}{dt dV d\Omega dV}. \qquad (III.9)$$

We have also seen, in Sect. I.2.4, the emissivity:

$$\epsilon_v(v, \vec{r}, \theta, \phi) = 4\pi \frac{dE_{\text{em}}}{dt dmd\Omega dV} = 4\pi \frac{dE_{\text{em}}}{dt \rho dV d\Omega dV}, \qquad (III.10)$$

where $\rho$ is the mass density of the ISM, and $dm = \rho dV$, its mass element. The factor $4\pi$, in Eq. (III.10), makes $\epsilon_v$ the solid angle fraction of monochromatic power emitted in a given direction, per unit mass. In Sect. I.2.4, we were considering the volume and mass of a grain, whereas, here, we are considering the volume and mass of the ISM. Combining Eq. (III.9) and Eq. (III.10), we get:

$$j_v(v, \vec{r}, \theta, \phi) = \frac{\rho(\vec{r})}{4\pi} \epsilon_v(v, \vec{r}, \theta, \phi). \qquad (III.11)$$

<figure><img src="image_253.png" /></figure>
<figure><img src="image_254.png" /></figure>

<figure><img src="image_255.png" /></figure>

<figure><img src="image_256.png" /></figure>

<figure><img src="image_257.png" /></figure>

FIGURE III.1 – Moments of the specific intensity. Panel (a) represents the specific intensity (Eq. III.1). It is the monochromatic power of light rays per area, dA, and within the solid angle dΩ. Panel (b) demonstrates the calculation of the flux (Eq. III.4). We have represented several rays, with different directions and intensities. The effective area perpendicular to rays going through dA, but that are inclined at an angle θ, is only cos θ dA. Panel (c) represents the radiation pressure (Eq. III.5). It is similar to panel (b), except that we have shown the component of the momentum vector, $\vec{p}$, perpendicular to the surface, $|\vec{p}| \cos \theta$. Only this component contributes to the pressure on dA. Panel (d) represents the energy density (Eq. III.6). Between times t and t+dt, photons going through dA are encompassed within the cylinder of volume cdtdA. Licensed under CC BY-SA 4.0.

**Extinction coefficient and opacity.** The amount of specific intensity absorbed and scattered along an infinitesimal path length, dl, in the direction (θ, φ), is the extinction coefficient, α ≥ 0, defined such that:

$$ \frac{dI_v(v, \vec{r}, \theta, \phi)}{dl} = -\alpha(v, \vec{r}) I_v(v, \vec{r}, \theta, \phi). \qquad (III.12) $$

Similarly to the convention we have adopted for κ in Sect. I.2.4, we pose α ≡ α<sub>ext</sub> = α<sub>abs</sub> + α<sub>sca</sub> to distinguish absorption and scattering. Eq. (III.12) can be expressed with microscopic quantities, assuming the absorbers and scatterers have a cross-section C<sub>ext</sub>(v, r) and a density n(r):

$$ \alpha(v, \vec{r}) = n(\vec{r}) C_{\text{ext}}(v, \vec{r}). \qquad (III.13) $$

If the composition of the ISM is homogeneous, then C<sub>ext</sub>(v, r) = C<sub>ext</sub>(v). The opacity, that we have seen in Sect. I.2.4, is related to α by:

$$ \alpha(v, \vec{r}) = \rho(\vec{r}) \kappa(v, \vec{r}). \qquad (III.14) $$

**Mean free path.** The mean free path of a photon with frequency, v, at position r can be defined as:

$$ l_{\text{mean}}(v, \vec{r}) \equiv \frac{1}{\alpha(v, \vec{r})} = \frac{1}{\rho(\vec{r})\kappa(v, \vec{r})}. \qquad (III.15) $$

It is the average length a photon will be able to travel before being absorbed or scattered. We can make the same remark as for the emissivity: in Sect. I.2.4, we were considering the cross-section of

<figure><img src="image_258.png" /></figure>
<figure><img src="image_259.png" /></figure>

dust particles per mass of grain, whereas Eq. (III.14) gives the cross-section of the whole ISM per mass of ISM. Table III.1 gives typical values of $l_{\text{mean}}$ in a homogeneous medium, assuming the dust constitution of the THEMIS model (cf. Sect. II.3.2.2).

In the diffuse ISM (WNM; $n_{\text{H}} \approx 0.3 \text{ cm}^{-3}$; Table III.6), the mean free path of a photon in the visible range is of the order of a kiloparsec.

<table><caption>TABLE III.1 – Mean free path as a function of wavelength and density. These quantities were computed by rewriting Eq. (III.15) as $1/l_{\text{mean}}(\lambda) = n_{\text{H}}m_{\text{H}}Y_{\text{dust}}\kappa(\lambda)$, taking $Y_{\text{dust}} = M_{\text{dust}}/M_{\text{H}}$ from Table II.4 and $\kappa$ values from Table II.5. We quote $l_{\text{mean}}$ at the same photometric bands as in Table II.5. These values correspond to Solar metallicity, $Z = Z_{\odot}$. At first order, for metallicities $Z \gtrsim 0.2 Z_{\odot}$, we can assume linearity: $l_{\text{mean}}(Z) \approx l_{\text{mean}}(Z_{\odot}) \times Z_{\odot}/Z$ (cf. Chap. IV). The different ISM phases quoted here (HIM, WNM, CNM) will be defined in detail in Sect. III.3.1.</caption><thead><tr><th rowspan="2"></th><th>HIM</th><th>WNM</th><th>CNM</th><th colspan="2">Molecular clouds</th></tr><tr><th>$n_{\text{H}} = 0.003 \text{ cm}^{-3}$</th><th>$n_{\text{H}} = 0.3 \text{ cm}^{-3}$</th><th>$n_{\text{H}} = 30 \text{ cm}^{-3}$</th><th>$n_{\text{H}} = 10^4 \text{ cm}^{-3}$</th><th>$n_{\text{H}} = 10^6 \text{ cm}^{-3}$</th></tr></thead><tbody><tr><td>$l_{\text{mean}}(\text{U})$</td><td>139 kpc</td><td>1.39 kpc</td><td>13.9 pc</td><td>0.0417 pc</td><td>86.1 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{B})$</td><td>177 kpc</td><td>1.77 kpc</td><td>17.7 pc</td><td>0.0532 pc</td><td>110 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{V})$</td><td>223 kpc</td><td>2.23 kpc</td><td>22.3 pc</td><td>0.0669 pc</td><td>138 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{R})$</td><td>275 kpc</td><td>2.75 kpc</td><td>27.5 pc</td><td>0.0824 pc</td><td>170 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{I})$</td><td>358 pc</td><td>3.58 kpc</td><td>35.8 pc</td><td>0.107 pc</td><td>222 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{J})$</td><td>691 kpc</td><td>6.91 kpc</td><td>69.1 pc</td><td>0.207 pc</td><td>427 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{H})$</td><td>1021 kpc</td><td>10.2 kpc</td><td>102 pc</td><td>0.306 pc</td><td>632 a.u.</td></tr><tr><td>$l_{\text{mean}}(\text{K})$</td><td>1734 kpc</td><td>17.3 kpc</td><td>173 pc</td><td>0.52 pc</td><td>1073 a.u.</td></tr></tbody></table>

### III.1.1.2 The Radiative Transfer Equation

The radiative transfer equation accounts for the variation of the specific intensity under the effects of absorption, scattering and emission (cf. Steinacker et al., 2013, for a review in the case of a dusty medium). It is schematically represented in Fig. III.2. This equation can be written:

$$ \frac{dI_{\text{V}}(\mathbf{v}, \vec{r}, \theta, \phi)}{dl} = -\underbrace{\alpha_{\text{abs}}(\mathbf{v}, \vec{r})I_{\text{V}}(\mathbf{v}, \vec{r}, \theta, \phi)}_{\text{absorption}} - \underbrace{\alpha_{\text{sca}}(\mathbf{v}, \vec{r})I_{\text{V}}(\mathbf{v}, \vec{r}, \theta, \phi)}_{\text{scattering out of the sightline}} + \underbrace{\alpha_{\text{sca}}(\mathbf{v}, \vec{r})2\pi \int_{-1}^{1} \Phi(\cos\theta', \mathbf{v})I_{\text{V}}(\mathbf{v}, \vec{r}, \theta(\theta'), \phi(\theta')) d\cos\theta'}_{\text{scattering in the sightline}} + \underbrace{j_{\text{V}}^{\text{dust}}(\mathbf{v}, \vec{r})}_{\text{dust emission}} + \underbrace{j_{\text{V}}^{*}(\mathbf{v}, \vec{r})}_{\text{stellar emission}}. \quad (\text{III.16}) $$

**Scattering in the sightline:** this term is the integral of $\alpha_{\text{sca}}I_{\text{V}}$ (i.e. the scattered intensity) over the phase function, $\Phi$ (Eq. I.41). This expression depends on the relative angle, $\theta'$, between the incident rays and the scattered direction ($\theta, \phi$). This is the term that makes this equation an integro-differential equation, coupling all the directions together. This is why, numerical methods are required to solve Eq. (III.16). If we assume isotropic scattering (i.e. $\langle \cos\theta \rangle = 0$, corresponding to the Rayleigh regime; cf. Sect. I.2.2.3), this term simplifies and becomes: $\alpha_{\text{sca}}(\mathbf{v}, \vec{r})J_{\text{V}}(\mathbf{v}, \vec{r})$.

**Emission:** we have assumed that the dust and stellar emissions are both isotropic, (i.e. independent of $\theta$ and $\phi$). This is a reasonable assumption in the ISM. If we also assume that there is only one grain species, and that this species is at equilibrium with the radiation field, we can explicit: $j_{\text{V}}^{\text{dust}}(\mathbf{v}) = \alpha(\mathbf{v})B_{\text{V}}(\mathbf{v}, T_{\text{eq}})$. This simplification still contains the problem that to determine the equilibrium temperature, $T_{\text{eq}}$, we need to integrate $\alpha_{\text{abs}}I_{\text{V}}$ over all directions, and all frequencies.

<figure><img src="image_260.png" /></figure>
<figure><img src="image_261.png" /></figure>

Eq. (III.16) can be rewritten, using the optical depth, $\tau$, defined such that $d\tau(v) = \alpha(v, l) dl$, or:

$$ \tau(v, l) = \int_{0}^{l} \rho(l') \kappa(v, l') dl'. \qquad (III.17) $$

Eq. (III.15) implies that $\tau(l_{\text{mean}}) = 1$. At a given wavelength, a medium is: (i) optically thin or transparent, if $\tau \ll 1$; and (ii) optically thick or opaque, if $\tau \gg 1$. Replacing $dl$ by $d\tau/\alpha$ in Eq. (III.16), we obtain:

$$ \frac{dI_{\nu}(v, \vec{r}, \theta, \phi)}{d\tau} = -I_{\nu}(v, \vec{r}, \theta, \phi) + S_{\nu}(v, \vec{r}, \theta, \phi), \qquad (III.18) $$

where the source function, $S_\nu$, includes all the terms added to the specific intensity:

$$ S_{\nu}(v, \vec{r}, \theta, \phi) = \tilde{\omega}(v, \vec{r}) 2\pi \int_{-1}^{1} \Phi(\cos\theta', v) I_{\nu}(v, \vec{r}, \theta(\theta'), \phi(\theta')) d\cos\theta' + \frac{j_{\nu}^{\text{dust}}(v, \vec{r}) + j_{\nu}^{*}(v, \vec{r})}{\alpha(v, \vec{r})}. \qquad (III.19) $$

We will discuss exact numerical solutions to this equation in Sect. III.1.1.4. For now, we discuss trivial solutions, when some processes are assumed negligible.

<figure><img src="image_262.png" /><figcaption>FIGURE III.2 – The radiative transfer equation. This figure represents only the different processes contributing to the variation of the specific intensity at one location, $\vec{r}$, in one direction, $(\theta, \phi)$. The blue sphere represents a volume element. The red spheres within represent dust grain and the cyan star represents an actual star that would be present in the volume element. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Propagation in vacuum.** Let's assume we have a star of radius, R, and surface temperature, $T_\star$, located at $\vec{r} = \vec{0}$. The flux at $r = R$ has to be integrated only over the hemisphere where a surface element of the star emits: $F_\nu(R) = 2\pi \int_0^1 B_\nu(T_\star) \cos\theta d\cos\theta = \pi B_\nu(T_\star)$. If there is no ISM around the star, Eq. (III.16) simply becomes $dI_\nu/dl = 0$. The solution is thus $I_\nu = B_\nu(T_\star)$ along the directions coming from the star and 0 in all other directions. At an arbitrary distance, $r$, from the star, the solid angle it occupies is $\Omega_\star = R^2/r^2$. The flux is thus $F_\nu(r) = \pi B_\nu(T_\star)R^2/r^2$, which is the classic $1/r^2$ dilution of the flux.

**Emission only.** Let's assume we are observing, at submm wavelengths, a molecular cloud constituted of equilibrium grains at $T \approx 10$ K, with opacity $\kappa$. At these wavelengths, the extinction is negligible. Eq. (III.16) is therefore simply $dI_\nu/dl = \rho\kappa B_\nu(T)$ within the cloud. The solution is thus $I_\nu(l) = \kappa B_\nu(T) \int_{l_0}^l \rho(l) dl$, where $l_0$ is the position of the edge of the cloud along the direction of the sightline. If the density is constant, we get $I_\nu(l) = \rho\kappa B_\nu(T) \times (l - l_0)$, which can be simplified as $I_\nu(l) = \tau(l)B_\nu(T)$. This expression of the brightness is often used in radio-astronomy.

**Absorption only.** Let's assume we are observing a background star through a cold molecular cloud, in the MIR. At these wavelengths, the albedo is close to 0 (Sect. I.2.2.3). If we make the assumption that the background star is much brighter than the thermal emission of the cloud, Eq. (III.16) simply becomes $dI_\nu/dl = -\alpha I_\nu$. The solution is therefore $I_\nu(l) = I_\nu^* \exp[-\tau(l)]$ along the direction coming from the star and 0 in all other directions.

<figure><img src="image_263.png" /></figure>
<figure><img src="image_264.png" /></figure>

**Emission and absorption.** We can merge the two previous cases. It could correspond to a hot molecular cloud, observed at MIR wavelengths. Its thermal emission is absorbed by the cloud itself. The solution of Eq. (III.18), in this case, is:

$$I_v(\tau) = I_v^* \exp(-\tau) + \int_0^\tau \exp(\tau' - \tau) S_v(\tau') d\tau'. \qquad (III.20)$$

If the cloud contains a single grain species at temperature T, the solution becomes:

$$I_v(\tau) = B_v(T) + [I_v^* - B_v(T)] \times \exp(-\tau). \qquad (III.21)$$

If we look in a direction away from the background star, we get the classical self-absorption formula:

$$I_v(\tau) = B_v(T) \times [1 - \exp(-\tau)]. \qquad (III.22)$$

This solution is displayed in Fig. III.3. Eq. (III.22) has the following two limit regimes.

**Optically thin:** if $\tau \ll 1$, $I_v \approx \tau B_v(T)$, which is the “emission only” solution. The emission from the cloud is a grey body (Sect. I.2.4).

**Optically thick:** if $\tau \gg 1$, $I_v \approx B_v(T)$, which means the cloud is not anymore a grey body, but a perfect one.

<figure><img src="image_265.png" /><figcaption>FIGURE III.3 – Solution to the radiative transfer equation for an isothermal cloud, without scattering. We have assumed that we are looking at a background star ($T_{\star} = 6000$ K), through an homogeneous cloud of grains at equilibrium temperature, $T = 100$ K, with the THEMIS opacity. We have diluted the stellar emission by a factor $10^{-6}$, which could correspond to a situation where the actual angular area of the star is $10^{-6}$ times the beam of the telescope. We show the SED according to Eq. (III.22), varying the optical depth in the V band, $\tau_v$. We have included very large $\tau_v$, up to $10^5$, in order to demonstrate the asymptotic behavior of the dust emission tending toward a perfect black body. In typical ISM studies, it is however rare to find $\tau_v$ values higher than $\approx 100$. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Scattering and absorption with central illumination.** Let's assume that we have a homogeneous spherical cloud, of radius $R_{cl}$, and a central isotropically illuminating source, with monochromatic

<figure><img src="image_266.png" /></figure>
<figure><img src="image_267.png" /></figure>

luminosity, $L_v^*(\lambda)$. If $L_v^{esc}(\lambda)$ is the monochromatic luminosity escaping the cloud, the *escape fraction* can be defined as:

$$P_v^{esc}(\lambda) = \frac{L_v^{esc}(\lambda)}{L_v^*(\lambda)} = \exp[-\tau_{eff}(\lambda)]. \qquad (III.23)$$

The second equality defines $\tau_{eff}$ as the *effective optical depth* of the medium.

**Optically thick:** In the case of pure isotropic scattering ($g = \langle\cos\theta\rangle \approx 0$), it can be shown that the *net displacement* of a photon after $N_{sca}$ interactions is: $l_{eff} = \sqrt{N_{sca}}l_{mean}$ (cf. Chap. 1 of Rybicky & Lightman, 1979). The probability a photon will be absorbed, at the end of a free path, is $1 - \tilde{\omega}$. The mean number of free paths can thus be estimated by $N_{sca}(1 - \tilde{\omega}) = 1$, or: $N_{sca} = 1/(1 - \tilde{\omega})$. The optical depth of the cloud is $\tau \approx R_{cl}/l_{mean}$ (Eq. III.15 and Eq. III.17). The same way, the effective optical depth can be written $\tau_{eff} \approx R_{cl}/l_{eff}$. We thus get:

$$\tau_{eff} \approx \sqrt{1 - \tilde{\omega}\tau}. \qquad (III.24)$$

**Optically thin:** in this case, there is a low probability of interaction. Eq. (III.23) tells us that $\tau_{eff}$ accounts only for photons that have been absorbed. We therefore simply have (cf. Chap. 1 of Rybicky & Lightman, 1979):

$$\tau_{eff} = (1 - \tilde{\omega})\tau. \qquad (III.25)$$

This formula simply subtracts among the few photons that may have interacted with the grains those which have been scattered. Contrary to Eq. (III.24), it is valid for any value of g.

Városi & Dwek (1999, hereafter VD99) have proposed an empirical approximation to interpolate the two regimes of Eq. (III.24) and Eq. (III.25), resulting in the following escape fraction:

$$P_{esc}^{cen}(\lambda) \approx \exp \left[ -(1 - \tilde{\omega}(\lambda))^{\chi(\lambda)} \tau(\lambda) \right], \qquad (III.26)$$

with:

$$\chi(\lambda) \equiv 1 - \frac{1}{2} \left[ 1 - \exp\left(-\frac{\tau(\lambda)}{2}\right) \right] \sqrt{1 - g(\lambda)}. \qquad (III.27)$$

They have benchmarked this approximation with a Monte-Carlo radiative transfer model (cf. Sect. III.1.1.4). Both are in very good agreement in most of the astrophysically relevant parameter space. This solution is displayed in Fig. III.4.a.

**Scattering and absorption with uniform illumination.** VD99 have also proposed an approximation in the case where the internal illumination of the cloud is uniform. They start from the escape probability of a homogeneous sphere with uniform illumination, without scattering, given by Osterbrock (1989):

$$P_{esc}^{nosca} = \frac{3}{4\tau} \left[ 1 - \frac{1}{2\tau^2} + \left(\frac{1}{\tau} + \frac{1}{2\tau^2}\right) \exp(-2\tau) \right]. \qquad (III.28)$$

A demonstration of this formula is given in Appendix C of VD99. VD99 find, with a recursive argument, that the following expression is in relatively good agreement with a Monte-Carlo radiative transfer model:

$$P_{esc}^{uni}(\lambda) \approx \frac{P_{esc}^{nosca}(\lambda)}{1 - \tilde{\omega}(\lambda) [1 - P_{esc}^{nosca}(\lambda)]}. \qquad (III.29)$$

VD99 show that Eq. (III.29) perfectly agrees with the exact solution for a particular value of $g(\tau)$. The largest discrepancies, of about 20%, are found at high optical depth, for $g$ close to 0. Eq. (III.29) is demonstrated in Fig. III.4.b. We can see that, at a given $\tau_v$, this geometry has an overall larger escape fraction than central illumination.

<figure><img src="image_268.png" /></figure>
<figure><img src="image_269.png" /></figure>

**FIGURE III.4 – Escaping radiation from a spherical cloud.** In the three panels, we show the escaping SED of a homogeneous spherical cloud, made of THEMIS grains. We vary the V-band optical depth, $\tau_V$. The illuminating source is a star ($T_\star = 3 \times 10^4$ K), with a bolometric luminosity $L_\star = 10^4 L_\odot$. In panel (a), we show the case of central illumination. All the power is in the central source. Panel (b) shows the case of uniform illumination. The total power of the sources within the cloud is $L_\star$. Panel (c) shows external illumination. In this case, the flux at the surface of the cloud is $L_\star/4\pi R_{cl}^2$. Licensed under CC BY-SA 4.0.
**Scattering and absorption with external illumination.** There are also expressions in the case of a homogeneous spherical cloud, externally illuminated by an isotropic radiation field. VD99 derive the following approximation for the *absorbed fraction*:

$$P_{\text{abs}}^{\text{ext}}(\lambda) \equiv 1 - P_{\text{esc}}^{\text{ext}}(\lambda) \approx \frac{4\tau(\lambda)[1 - \tilde{\omega}(\lambda)]}{3} P_{\text{esc}}^{\text{uni}}(\lambda). \qquad (\text{III.30})$$

The absorbed fraction is indeed more relevant in the case of external illumination, as it would be difficult to observationally separate the escaping radiation from the cloud and the ambient ISRF. On the contrary the absorbed fraction is meaningful if one wants to evaluate the heating of the cloud. This approximation is demonstrated in Fig. III.4.c.

The three formulae for spherical clouds, given in Eq. (III.26), Eq. (III.29) and Eq. (III.30), do not allow us to model the internal heating of the cloud. These expressions are indeed global escape and absorbed fractions, but they do not account for the gradient of illumination within the cloud that would lead to a gradient of heating rate.

### III.1.1.3 Approximations for Clumpy Media

The ISM is a highly heterogeneous medium, with contrast densities of several orders of magnitude. A useful approximation is the *clumpy medium*, composed of: (i) a diffuse, uniform *InterClump Medium* (ICM), characterized by its density, $n_{\text{ICM}}$; and (ii) dense, spherical clumps, with density, $n_{\text{C}}$, radius, $r_{\text{C}}$, and volume filling factor, $f_{\text{v}}$.

**Effective optical depth of a clumpy medium.** Let's assume that we are looking at a background star through a cloud, and that the albedo of the grains is negligible. Along a given sightline, we have seen in Sect. III.1.1.2 that the brightness in the direction of the star is $I_{\text{v}} = I_{\text{v}}^{\star} \times \exp(-\tau)$, If the clumpy structure of the cloud is unresolved, the brightness measured in the telescope beam can be written as the sum of N sightlines, some passing through the ICM, others through the clumps:

$$I_{\text{v}}^{\text{clumpy}} = \frac{I_{\text{v}}^{\star}}{N} \sum_{i=1}^{N} \exp(-\tau_{i}). \qquad (\text{III.31})$$

This is the general expression. In the case of an homogeneous medium, Eq. (III.31) simplifies: $I_{\text{v}} = I_{\text{v}}^{\star} \exp(-\tau_{\text{hom}})$, where $\tau_{\text{hom}}$ is the optical depth of the homogeneous medium. In order to have the same dust mass and opacity as in the clumpy medium, we need to have:

$$\tau_{\text{hom}} = \frac{1}{N} \sum_{i=1}^{N} \tau_{i}. \qquad (\text{III.32})$$

In the homogeneous medium, $\rho \times L$ is indeed the mass surface density of a cloud of depth L. If there is a statistical distribution of clumps in the beam, it must be identical in the clumpy medium. Thus, the brightness of the homogeneous medium is:

$$I_{\text{v}}^{\text{hom}} = \exp\left(-\frac{1}{N} \sum_{i=1}^{N} \tau_{i}\right) = \left[ \prod_{i=1}^{N} \exp(-\tau_{i}) \right]^{1/N}. \qquad (\text{III.33})$$

Invoking the *arithmetic-mean/geometric-mean inequality*<fnref n="1" /> (e.g. page 456 of Cauchy, 1821), we conclude that: $I_{\text{v}}^{\text{clumpy}} \ge I_{\text{v}}^{\text{hom}}$. An important consequence of this result is that, from an observational point of view, we can miss a large mass of dust hidden in clumps. VD99 discuss this result in more detail.

The effective optical depth of a clumpy medium is always lower than that of a homogeneous medium with the same dust constitution and dust mass.

<footnote n="1">1. Stating that $\sum_{i=1}^{N} a_i / N \ge \prod_{i=1}^{N} a_i^{1/N}$, provided that $a_i \ge 0, \forall i$.</footnote>
<figure><img src="image_270.png" /></figure>

**The mega-grains approximation.** Neufeld (1991) proposed a simple approach to explain the leakage of Ly-α photons by galaxies. He treated dusty gas clumps, in an empty ICM, as large grains with their own albedo and asymmetry parameter. This idea was then further developed by Hobson & Padman (1993), in cases where the ICM is non empty. They named it the *mega-grains approximation*, as clumps are treated as grains, although they have macroscopic sizes. They applied this approach to a clumpy infinite slab, externally illuminated on one side, and compared the results to a Monte-Carlo radiative transfer model. VD99 then refined some of the expressions of Hobson & Padman (1993) and applied them to the case of a spherical clumpy cloud, with the three types of illuminations we have discussed in Sect. III.1.1.2: (i) central; (ii) uniform; and (iii) external. VD99 systematically benchmarked their results with a Monte-Carlo radiative transfer code. We briefly review their results in the rest of this section.

**Escape fractions for a clumpy medium.** VD99 derived a series of expressions, based on Eq. (III.26), Eq. (III.29) and Eq. (III.30), but replacing grain properties by effective mega-grains properties. These analytical approximations are all summarized in Sect. 5 of VD99. We demonstrate these analytical expressions for the three types of illuminations in Fig. III.5. Overall, they provide a good agreement with Monte-Carlo radiative transfer calculations. They are also very easy to compute. The weakest point concerns the treatment of the grain heating. The mega-grains formalism allows us to separate the absorbed fractions in the clumps and in the ICM. It thus provides different heating rates in the two phases. In Fig. III.5, we can clearly see that the clump emission (red) is significantly colder than that of the ICM (magenta). It however does not allow us to estimate the gradient of radiation field within the ICM and within clumps. This is the most dramatic in the case of central illumination. In order to obtain a more realistic SED for this particular case, VD99 used an *ad hoc* prescription, assuming a power-law distribution of equilibrium grain temperatures controlled by several tuning parameters depending on the grain type.

### III.1.1.4 Rigorous Solutions

The radiative transfer equation (Eq. III.16) can be solved numerically. There are two main classes of methods (Steinacker et al., 2013, for a review).

**Monte-Carlo Radiative Transfer (MCRT)** consists in simulating the random walk of photons, from their sources (stars, AGNs or thermal emission from dust grains) to the outside of the region, through their multiple scatterings.

**Ray-tracing** numerically solves the radiative transfer equation on a discretized grid along multiple sightlines. It is more difficult to implement than MCRT, and can be computationally more intensive. It however allows the user a better assessment of the numerical errors.

MCRT is by far the most popular method. In this section, we briefly review its principle and apply it to an example.

**Setting the model.** To solve the radiative transfer equation (Eq. III.16), we need to specify the following physical ingredients.

**A 3D spatial grid of dust density** has to be defined. Different coordinate systems can be chosen. In case there are steep density gradients (e.g. clumps), one has to make sure the transitions are finely enough sampled.

**A 3D distribution of primary emitters,** as well as their SED needs to be defined. In this section, we will consider only stars, but AGNs can be treated the same way. It is possible to account both for: (i) discrete emission, such as an individual star or a cluster; and (ii) diffuse emission, such as unresolved stellar distribution.

**The dust properties** need to include, at least: the opacity, the albedo and the asymmetry parameter. It is possible to account for a mixture of dust grains, with different compositions and sizes. The

<figure><img src="image_271.png" /></figure>
<figure><img src="image_272.png" /></figure>

<figure><img src="image_273.png" /><figcaption>FIGURE III.5 – Escaping SED from clumpy spherical clouds. The three panels are the SED of a $R_{cl}$ = 1 pc cloud, containing $r_C$ = 0.05 pc clumps, computed with the mega-grains approximation. The density is $n_{ICM}$ = 1000 cm$^{-3}$ in the ICM and $n_C$ = 10$^5$ cm$^{-3}$ in the clumps (volume filling factor, $f_v$ = 20%). The grains have THEMIS optical properties and are assumed to be at thermal equilibrium. We have displayed: (i) the intrinsic stellar luminosity ($T_★$ = $1.5 \times 10^4$ K; $L_★$ = $3.3 \times 10^4$ L$_⊙$), in grey; (ii) the escaping stellar radiation, in yellow; (iii) the clump luminosity, in red; (iv) the ICM luminosity, in magenta; (v) the total escaping SED, in cyan. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_274.png" /></figure>
<figure><img src="image_275.png" /></figure>

size-distribution-integrated quantities of Eqs. (II.19) – (II.22) are used for the transfer of stellar photons. It is however necessary to treat the thermal emission of individual species, especially if they are stochastically heated.

**The principle of Monte-Carlo radiative transfer.** To compute a MCRT model, we need to draw a large number of photons (typically ≈ 10⁶ per wavelength bin), and execute the following steps. The procedure is schematically represented on Fig. III.6.

1. At each wavelength, photons are randomly drawn from primary emitters, proportionally to their specific intensity. The emission angle, (θ, φ), is randomly chosen, if the emitters are isotropic, which is the case for stars.

2. The interaction probability of a photon with a dust grain is then drawn from 1 − exp(−τ), along the original direction of the photon. On interaction, one accounts for the probabilities of: (i) absorption (proportional to 1 − ˜ω); and (ii) scattering (proportional to ˜ω). We could randomly choose between absorption and scattering, and thus terminate the path of the photons 1 − ˜ω of the times. However, forced scattering, the way it is shown in Fig. III.6, allows us to track the absorption and scattering probabilities at each interaction, in a numerically more efficient way. The scattering angle is randomly drawn from the scattering phase function (Eq. I.44), setting the new direction of the photon.

3. We iterate this process as long as needed, until the photon exits the nebula. The average number of interactions increases with the effective optical depth of the nebula. At each interaction, $(1-\tilde{\omega}) \times I_v$ is absorbed by the grain, and $\tilde{\omega} \times I_v$ is scattered. After N interactions, the scattered intensity is $\tilde{\omega}^N \times I_v$ and the absorbed power is proportional to $(1-\tilde{\omega}) \times \tilde{\omega}^{N-1}$. For a typical $\tilde{\omega} \approx 0.5$, after three scatterings, the intensity is reduced by a factor $\approx 0.13$.

4. Once all the photons at all wavelengths have been drawn, the thermal emission of all dust species within each cell can be computed. IR photons are then drawn and scattered through the nebula, the same way as stellar photons. The overall opacity and albedo are usually much lower in the IR. The computation of IR radiative transfer is thus usually much faster. In the most embedded regions, the IR radiation absorbed by dust grains can be significant. One therefore has to recompute the IR transfer a few times, until an energy balance is reached.

These steps constitute the most basic implementation of MCRT. Numerous optimizations have how-
ever been proposed in the last fifty years (e.g. Witt, 1977a,b,c; Witt & Oshel, 1977; Yusef-Zadeh et al.,
1984; Whitney & Hartmann, 1992; Wood, 1997; Wood & Jones, 1997; Városi & Dwek, 1999; Baes & De-
jonghe, 2001; Gordon et al., 2001; Misselt et al., 2001; Steinacker et al., 2002, 2006; Wood et al., 2008;
Baes et al., 2011; Camps & Baes, 2015; Siebenmorgen et al., 2015; Natale et al., 2015; Juvela, 2019).
Improvements and optimizations include: (i) massive parallelization, and the use of GPU; (ii) the
production of synthetic photometric images; (iii) the treatment of polarization by scattering.

**Numerical method to randomly draw photons.** To simulate the random walk of a photon, there are two sets of random variables to draw: (i) random interaction events; and (ii) scattering angles.

**Drawing interaction events** along the path of a photon consists in drawing a path length from the following *Probability Density Function* (PDF):

$$
p(l) = 1 - \exp \left( - \int_{0}^{l} \kappa \rho(l') l' dl' \right). \qquad (\text{III.34})
$$

For simplicity, *l* corresponds to the path length along the direction of the photon, starting at *l* = 0. Drawing a variable from this distribution can be achieved the following way (e.g. Városi & Dwek, 1999). It uses the *rejection method* (cf. Appendix C.2.3.1). These steps are represented on Fig. III.7.

1. First draw a uniform random variable between 0 and 1, Θ₁.

<figure><img src="image_276.png" /></figure>
<figure><img src="image_277.png" /><figcaption>FIGURE III.6 – Principle of Monte-Carlo radiative transfer. The blue area represents a dust cloud. We follow the journey of a single photon emitted by a star into the cloud and scattered at different locations. A single photon will be either scattered or absorbed. However, for numerical efficiency, we consider both solutions, weighted by their probability (scattering: $\tilde{\omega}$; absorption: $1 - \tilde{\omega}$). The dark red arrows correspond to emitted IR photons. In principle, these photons can also be scattered and absorbed by the cloud. The arrows exiting the cloud to the right represent what would be measured by an observer. Licensed under CC BY-SA 4.0.</figcaption></figure>

2. Set $l_0 = -\ln(1 - \Theta_1)/\kappa\rho_{max}$, where $\rho_{max}$ is the maximum of the density along the path.

3. Draw a second uniform random variable between 0 and 1, $\Theta_2$.

4. If $\Theta_2 \le \rho(l_0)/\rho_{max}$, then the interaction is accepted, and a new scattering angle can be drawn. On the contrary, if $\Theta_2 > \rho(l_0)/\rho_{max}$, the interaction is rejected. One then needs to go back to the first step, starting from $l_0$, this time.

**Scattering angles** are drawn from the scattering phase function (Eq. I.44; Fig. III.8.a). If we place a spherical coordinate reference frame at the position of interaction with its z-axis aligned with the incoming direction of the photon, then the two spherical angle, $(\theta, \phi)$, that will determine the new direction of the photon are drawn the following way.

1. The new polar angle, $\theta$, is drawn by inverting the Cumulative Distribution Function (CDF) (cf. Appendix C.2.3.2). In case we use the Henyey & Greenstein (1941) phase function, the inverse of the CDF can be analytically derived (cf. Fig. III.8.b):

$$F^{-1}(\Theta_3) = \frac{1}{2g} \left[ 1 + g^2 - \left( \frac{1 - g^2}{1 - g + 2g\Theta_3} \right)^2 \right], \qquad (III.35)$$

where $\Theta_3$ is a uniform random variable between 0 and 1.

2. The new azimuthal angle, $\phi$, is simply drawn from a uniform distribution between 0 and $2\pi$, because there is a symmetry of revolution around the direction of the photon.

**Example of a clumpy medium.** For this manuscript, we have developed a MCRT model, following the procedure previously described. We have applied it to a spherical clumpy cloud, similar to those discussed in Sect. III.1.1.3. The radius is $R_s = 1$ pc, with an ICM density, $n_{ICM} = 1000 \text{ cm}^{-3}$. The $r_c = 0.05$ pc clumps have a density, $n_c = 10^5 \text{ cm}^{-3}$, with a filling factor, $f_v = 20\%$. We assume
<figure><img src="image_278.png" /></figure>

<figure><img src="image_279.png" /><figcaption>**FIGURE III.7 – Drawing photons in a Monte-Carlo radiative transfer model.** This figure shows the dust density of a medium (in blue; normalized to its peak value, $\rho_{max}$), as a function of the path length of the photon, $l$, across the nebula. The x-axis is multiplied by $\kappa\rho_{max}$ so that it corresponds to the optical depth the medium would have if the density was everywhere $\rho_{max}$. Two random draws are represented. In magenta, we show a first draw, bringing the photon to $l_1 = -\ln(1 - \Theta_1)/\kappa\rho_{max}$. The interaction at $l_1$ is rejected because $\Theta_2 > \rho(l_1)/\rho_{max}$. A second similar draw brings the photon at $l = l_1 + l_2$, where the interaction is accepted. After this draw, one needs to draw the scattering angle. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_280.png" /><figcaption>**FIGURE III.8 – Drawing scattering angles in a Monte-Carlo radiative transfer model.** Panel (a) shows the Henyey & Greenstein (1941) scattering phase function (Eq. I.44), for different values of the asymmetry parameter, $g = \langle\cos\theta\rangle$. Panel (b) shows the inverse CDF (Eq. III.35), for different values of $g$. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_281.png" /></figure>
THEMIS grain constitution, and assume all grains are at thermal equilibrium. This cloud is centrally illuminated by a $T_{\star} = 1.5 \times 10^{4}$ K star, with $L_{\star} = 3.3 \times 10^{4}$ L$_{\odot}$.

**The projected map** of the dust mass surface density is shown in Fig. III.9.a. The average absorbed fraction, $\langle P_{\text{abs}} \rangle$ is shown in Fig. III.9.b. The latter quantity is the projected average of the absorbed fraction of photons passing through the cell. This fraction is highly concentrated in the central region, as most of the power is absorbed by clumps close to the star. Those clumps are hot, whereas clumps on the outskirt are colder, because they are essentially heated by fewer photons that have been multiply scattered. Fig. III.10 represents a few photon paths at three different wavelengths.

**The total SED** of this cloud is shown in Fig. III.11. We notice that a fraction of the dust emission is self-absorbed. This is mainly the 10 and 18 µm silicate emission of the hottest central clumps being absorbed by the outer ones. We have also compared our model to the mega-grains approximation (Sect. III.1.1.3; in dashed blue). We see this approximation is quite good in the UV-to-NIR range. It is however discrepant in the IR. This is because there is a very strong gradient of heating conditions, seen in Fig. III.9.b, whereas the mega-grains approximation accounts only for the total absorbed fraction in the clumps and the ICM$^2$.

Despite its intensive numerical requirements, MCRT is thus the most flexible and accurate way to compute the SED of an interstellar cloud. It is however important to make sure that the spatial resolution of the density grid is fine enough to resolve the shortest mean free path of photons. Otherwise, we would be smearing out a potential sub-grid temperature gradient. For the present model, we have 0.01 pc cells for $l_{\text{mean}}(\text{U}) \approx 0.005$ pc.

To accurately compute a radiative transfer model, it is necessary to resolve scale-lengths of the order of the mean free path of UV photons.

<figure><img src="image_282.png" /><figcaption>(b) Average absorbed power, $\langle P_{\text{abs}} \rangle$</figcaption></figure>

<figure><img src="image_283.png" /></figure>

**FIGURE III.9 – Spatial distributions of the clumpy radiative transfer model.** Panel (a) shows the projected dust mass surface density of the cloud. Panel (b) shows the average projected absorbed fraction, $\langle P_{\text{abs}} \rangle$. The star is located at the center of the cloud. Notice that most of the power is absorbed in the central region. Licensed under CC BY-SA 4.0.

<footnote n="2">2. VD99 solved this issue by assuming a power-law distribution of temperatures that we have discussed in Sect. III.1.1.3. This is however an *ad hoc* solution that needs to be calibrated for each dust species, by using an actual MCRT model. Our goal being to demonstrate what a MCRT model brings, we chose to compare it only to the SED that the mega-grains approximation allows us to derive, by itself.</footnote>
<figure><img src="image_284.png" /></figure>

<figure><img src="image_285.png" /></figure>

FIGURE III.10 – *Random photon path within a clumpy medium.* The three grey-scale images represent the projected column density of Fig. III.9.a. We have overlaid the random path of 10 photons before they exit the cloud, at three different wavelengths. These draws are those computed by the MCRT model. For a given wavelength, the different colors represent different photons. These figures demonstrate the fact that, at short wavelength, the mean free-path is shorter, resulting in a larger number of scatterings. Licensed under CC BY-SA 4.0.

<figure><img src="image_286.png" /></figure>

FIGURE III.11 – *Total SED from the clumpy radiative transfer model.* The yellow curve represents the intrinsic SED of the central star. The red curve is the intrinsic emission (non self-extincted) of the dust emission of the whole cloud. The grey curve shows the total escaping SED of the cloud. We have overlaid the mega-grains approximation (Sect. III.1.1.3), in dashed-blue, for comparison. Licensed under CC BY-SA 4.0.

<figure><img src="image_287.png" /></figure>
<figure><img src="image_288.png" /></figure>

### III.1.2 Approximate Treatments of the Mixing of Physical Conditions

Ideally, each time we study a Galactic region or a galaxy, we should solve the radiative transfer equation (Eq. III.16). This is however, most of the time, impossible, because of the lack of constraints on the actual 3D structure of the region. Even if we have a collection of high-angular resolution multiwavelength images of our object, the matter and stellar distributions along the sightline have to be inferred. This inference is possible when the large-scale geometry of the object is quite regular, for instance: a protostellar disk and its jet, or a galactic disk and its bulge, etc. We will discuss the MCRT modeling of disk galaxies in Sect. III.1.3.2. Otherwise, most often, we need to adopt empirical approaches that allows us to constrain the dust properties, despite our uncertainty of the spatial structure of the region.

#### III.1.2.1 The Historical Model: the MBB

The MBB (cf. Sect. I.2.4.1) is historically the most widely-used dust SED model. It is controlled by the three following parameters (Eq. I.71 and Eq. I.73).

1. The dust mass, $M_{\text{dust}}$, is a scaling parameter.

2. The equilibrium temperature, T, controls the emission peak wavelength.

3. The emissivity index, $\beta$ (Eq. I.73), controls the long-wavelength slope of the SED. This is demonstrated in Fig. III.12.a.

Its physical assumptions are simplistic: (i) the IR emission is optically thin; (ii) the dust is made of a single species of grains at thermal equilibrium with the ISRF; and (iii) the opacity is a power-law. By inferring both T and $\beta$, this model is designed to constrain both the dust excitation and its optical properties. This model was popularized by Hildebrand (1983), in the IRAS days (cf. Sect. II.1.2). At the time, it was well adapted, being a simple, but still physical model, with only three parameters to fit four broadbands (the four IRAS bands at 12, 25, 60 and 100 µm). It has however several limitations that are often disregarded in the literature.

**The mixing of physical conditions** is not accounted for by the MBB. It means that the SED fit of a complex region, containing a gradient of temperatures, is biased (e.g. Juvela & Ysard, 2012; Galliano et al., 2018). This is demonstrated in Fig. III.12.b. The grey-filled curve is the SED coming from a power-law distribution of temperatures (T = 15 to 60 K; the color curves). The intrinsic emissivity index of the grains making up this region is $\beta = 1.79$. Yet, fitting such a SED with a single MBB leads to a compromise temperature (T = 45 ± 8 K) and a systematically lower emissivity index, $\beta = 1.12 \pm 0.30$. This is because a gradient of temperature tends to flatten the FIR slope, similarly to a lower $\beta$ value.

The emissivity index derived from a single MBB fit is always lower than its true, intrinsic value.

**Stochastic heating** is formally equivalent to a gradient of temperatures (cf. Sect. I.2.4.3). Stochastically heated grains dominate the emission at MIR wavelengths (cf. Fig. II.27). A single MBB is thus biased at short wavelengths by small grains, the same way it is biased at long wavelengths by cold dust. Solving this issue by fitting a linear combination of two or three MBBs can palliate this problem, but is usually not sufficient. In addition, fitting an out-of-equilibrium emission with equilibrium grains is physically incorrect. It renders the interpretation of the parameters of the hot MBB unreliable.

**Realistic opacities are more complex** than a power-law. Laboratory data show that emissivity indices of actual materials are wavelength-dependent quantities, $\beta(\lambda)$ (cf. e.g. Fig. II.22). Additionally, the somehow arbitrary choice of the two other parameters in Eq. (I.73) – the reference wavelength, $\lambda_0$, and the scaling of the opacity at this wavelength, $\kappa_0$ – have dramatic consequences on the derived dust mass. It is thus important to limit the potential range of variations of these parameters. The Kramers-Kronig relations (Sect. I.2.1.6) impose that $\beta \ge 1$, and

<figure><img src="image_289.png" /></figure>
<figure><img src="image_290.png" /></figure>

most compounds studied in the laboratory have $\beta < 2.5$ (cf. e.g. the Jena database). We recommend calibrating $\kappa_0$ on laboratory data or on well-constrained dust models (such as Eq. II.26). Contrary to what Hildebrand (1983) recommended, it is probably better to choose a reference wavelength, $\lambda_0$, around the peak of the FIR SED (between 100 and 300 µm), rather than in the submm. This way, variations of $\beta$ will impact only moderately the derived dust mass.

☞ A MBB fit infers parameters whose physical meaning is difficult to assess.

**The noise-induced degeneracy** between T and $\beta$ is well-documented (e.g. Shetty et al., 2009; Kelly et al., 2012; Galliano, 2018). A false negative correlation arises when a series of MBB fits are performed with standard least-squares, maximum likelihood, or non-hierarchical Bayesian methods. This degeneracy prevents to explore the potential correlation of these parameters that laboratory data and solid-state models suggest (e.g. Mennella et al., 1998; Meny et al., 2007). Luckily, hierarchical Bayesian methods are efficient at removing false, noise-induced correlations (e.g. Kelly et al., 2012; Galliano, 2018). Chap. V is entirely devoted to fitting techniques.

<figure><img src="image_291.png" /></figure>

FIGURE III.12 – MBB fitting. Panel (a) demonstrate the effect of the emissivity index, $\beta$ (Eq. I.73), on the SED of a MBB. The temperature is T = 30 K, with $\kappa(\lambda_0 = 250 \mu m) = 0.64 m^2/kg$ (Eq. II.26). Panel (b) illustrates the limitation of the isothermal assumption. The black error bars are synthetic observations (noise has been added). They are sampling the grey-filled curve, which is the integral of MBBs (color curves) times a power-law distribution of temperatures (T_min = 15 K, T_max = 60 K, and index 4). These MBBs have THEMIS optical properties ($\beta = 1.79$; Eq. II.26). Yet the fitted value (black dashed curve) is significantly lower: $\beta = 1.12 \pm 0.30$. The fitted temperature falls in the middle of the range, toward the high end: T = 45 $\pm$ 8 K. Licensed under CC BY-SA 4.0.

### III.1.2.2 A Phenomenological, Composite Approach

Dust models provide useful frameworks to model SEDs (Sect. II.3). Without the possibility to compute the radiative transfer, we are however facing the problem of the mixing of physical conditions. A prescription, proposed by Dale et al. (2001), has proven to be a powerful solution to this issue. It consists in assuming that the dust mass is distributed in regions with different starlight intensities,

<figure><img src="image_292.png" /></figure>
<figure><img src="image_293.png" /></figure>

U, following a power-law:

$$ \frac{1}{M_{\text{dust}}} \frac{dM_{\text{dust}}}{dU} = U^{-\alpha} \times \left\{ \begin{array}{ll} \frac{1-\alpha}{(U_{-} + \Delta U)^{1-\alpha} - U_{-}^{1-\alpha}} & \text{if } \alpha \neq 1 \\ \\ \frac{1}{\ln(U_{-} + \Delta U) - \ln U_{-}} & \text{if } \alpha = 1 \end{array} \right\} \quad \text{for } U_{-} \le U \le U_{-} + \Delta U. \qquad (\text{III.36}) $$

The idea is that the shape of the observed SED is used to constrain this distribution of ISRFs, assuming a dust mixture constitution. By lack of a better term, we call this approach the composite model. The free parameters are:

* the total dust mass, $M_{\text{dust}}$, acting as a scaling parameter;

* the power-law index, $\alpha$;

* the minimum starlight intensity, $U_{-}$;

* the width of the starlight intensity distribution, $\Delta U$.

It thus provides a way to account for the potential complexity of the region without having to model the radiative transfer. The model SED is then simply:

$$ L_{\nu}^{\text{dust}}(\lambda) = \int_{U_{-}}^{U_{-} + \Delta U} \epsilon_{\nu}(\lambda, U) \frac{dM_{\text{dust}}}{dU} dU, \qquad (\text{III.37}) $$

where $\epsilon_{\nu}(\lambda, U)$ is the monochromatic emissivity of the dust model, exposed to a single starlight intensity, U (Fig. II.24). Fig. III.13.a shows an example of a SED fit, using Eq. (III.36). We have added a free-scaling black body, at $T_{\star} = 30,000$ K, to account for the stellar continuum that may be contaminating the MIR photometric bands. The composite approach is flexible enough to be usable in a diversity of environments. Dale et al. (2001) lists several simple geometries for which Eq. (III.36) is actually the solution. It is also adapted to more complex ISM topologies. For instance, Fig. III.13.b shows the dust mass distribution as a function of U, for each cell in the MCRT simulation of Sect. III.1.1.4. Despite the complex, clumpy structure of this cloud, it can be reasonably well-approximated by a power-law (shown in red).

**The average starlight intensity.** The parameters of Eq. (III.36) do not have a very clear physical meaning. Besides, they are often degenerate: the uncertainties on the three parameters $U_{-}$, $\Delta U$ and $\alpha$ are strongly correlated. It can be more efficient to quote the average of the distribution:

$$ \langle U \rangle = \frac{1}{M_{\text{dust}}} \int_{U_{-}}^{U_{-} + \Delta U} U \frac{dM_{\text{dust}}}{dU} dU = \begin{cases} \frac{1-\alpha}{2-\alpha} \times \frac{(U_{-} + \Delta U)^{2-\alpha} - U_{-}^{2-\alpha}}{(U_{-} + \Delta U)^{1-\alpha} - U_{-}^{1-\alpha}} & \text{if } \alpha \neq 1 \& \alpha \neq 2 \\ \\ \frac{\Delta U}{\ln(U_{-} + \Delta U) - \ln U_{-}} & \text{if } \alpha = 1 \\ \\ \frac{\ln(U_{-} + \Delta U) - \ln U_{-}}{U_{-}^{-1} - (U_{-} + \Delta U)^{-1}} & \text{if } \alpha = 2. \end{cases} \qquad (\text{III.38}) $$

This parameter quantifies the *average starlight intensity*, heating the bulk of the dust mass. It is the equivalent of the equilibrium temperature of a MBB, as it controls the peak emission wavelength, except that it accounts for the mixing of physical conditions and the stochastic heating of small grains. The Total InfraRed (TIR) luminosity, $L_{\text{TIR}}$, can be expressed:

$$ L_{\text{TIR}} = \int_{3 \text{ µm}}^{1000 \text{ µm}} L_{\lambda}(\lambda) d\lambda \approx \int_{0}^{\infty} L_{\nu}^{\text{dust}}(\nu) d\nu = \epsilon \times M_{\text{dust}} \times \langle U \rangle, \qquad (\text{III.39}) $$

where the constant $\epsilon \equiv \int \epsilon_{\nu} dv$ is the bolometric emissivity of the dust model. For the THEMIS mixture, heated by the Mathis et al. (1983) ISRF, it is $\epsilon = 221 L_{\odot}/M_{\odot}/U$ (Table II.7).

<figure><img src="image_294.png" /></figure>
<figure><img src="image_295.png" /></figure>

FIGURE III.13 – *Phenomenological mixing of physical conditions.* Panel (a) shows an example of SED fit using the composite approach (Eq. III.36). The black error bars represent synthetic observations (noise has been added). These observations have been fitted with the total model represented by the dark grey line. This total model is the sum of a stellar (in yellow) and a dust (in light grey) components. The dust component is the integral of dust models illuminated by different U (Eq. III.36). The rainbow curves represent several bins of ISRF. The sum of these rainbow curves is the light-grey-filled curve. The blue curve in panel (b) shows the distribution of dust mass per bin of ISRF intensity, U (Sect. I.2.4.2), for each cell in the MCRT simulation presented in Sect. III.1.1.4. The value of U has been derived from the grain equilibrium temperature, using Eq. (I.84). The red line shows a power-law approximation of this distribution. With the parametrization of Eq. (III.36), it would correspond to: $U_{-} = 0.03$, $\Delta U = 10^4$ and $\alpha = 0.5 + 1 = 1.5$ (the +1 comes from taking the derivative of $M_{dust}(U)$ to get Eq. III.36). Licensed under CC BY-SA 4.0.

**Constraining the dust properties.** The composite approach of Eq. (III.36) allows us to constrain parameters that are not extremely sensitive to radiative transfer effects (i.e. to variations of the local intensity and spectral shape of the ISRF; cf. e.g. Galliano et al., 2018, 2021, for a discussion).

**The dust mass, $M_{dust}$, is dominated by large grains (cf. Table II.3).** The heating of these large grains is sensitive only to the power they absorb, and not to the spectral shape of the ISRF, contrary to small grains (cf. Sect. I.2.4.3). The assumption of a constant ISRF shape therefore does not significantly bias $M_{dust}$.

**The average starlight intensity, $\langle U \rangle$ (Eq. III.38), is empirically constrained by the shape of the FIR emission, dominated by large grains.** This quantity is thus not particularly biased, similarly to $M_{dust}$. In addition, Eq. (III.39) tells us that $L_{TIR} \propto M_{dust} \langle U \rangle$. Yet, $L_{TIR}$ is a weakly model-dependent quantity, as it is simply the integral of the SED model passing through the observed fluxes. $L_{TIR}$ and $M_{dust}$ being reliable, $\langle U \rangle$ has to be.

**The mass fraction of PAH, $q_{PAH}$ (or the mass fraction of small a-C(:H), $q_{AF}$, for the THEMIS model),** discussed in Sect. II.3.3.1, controls the strength of the MIR aromatic features (cf. Sect. II.2.2.1). These features, being carried by small, stochastically heated grains, are sensitive to the spectral shape of the ISRF (cf. Fig. II.27.a). This is demonstrated in Fig. III.14. We have considered two extreme ISRFs (Fig. III.14.a). The bias on the aromatic feature emission is at most a factor $\approx 1.8$ (Fig. III.14.b).

☞ Eq. (III.36) provides acceptable estimates of $M_{dust}$, $\langle U \rangle$ and $q_{PAH}$ (or $q_{AF}$).
<figure><img src="image_296.png" /></figure>

<figure><img src="image_297.png" /><figcaption>FIGURE III.14 – Effect of the ISRF hardness on the SED. Panel (a) represents different ISRFs: (i) the Solar neighborhood (red; Mathis et al., 1983); and (ii) a B star (T<sub>★</sub> = 30 000 K; blue; Lanz & Hubeny, 2007). We have represented the spectral range of ionizing photons in yellow. Both ISRFs are normalized so that U = 1 (Eq. I.83). Panel (b) shows the SEDs of the THEMIS model, heated by the two ISRFs. We show the SED for different values of U, in order to obtain the same FIR SED, as this parameter is inferred by the fitter. The SEDs are normalized by the FIR luminosity, L<sub>FIR</sub>, between λ = 60 and 200 µm. The B star heating leads to an increase in the λ = 6 – 9 µm emission by a factor ≈ 1.8, compared to the Solar neighborhood. Licensed under CC BY-SA 4.0.</figcaption></figure>

**Limitations of the composite approach.** Using Eq. (III.36) to model typical broadband SEDs of Galactic regions and nearby galaxies is an efficient method. However, as any approximation, it has some limitations.

**Variation of the grain constitution** within the observed region is expected, as dust evolves with ISRF intensity and ISM density (cf. Chap. IV, which is devoted to dust evolution). The composite approach accounts for the fact that there are variations of the physical conditions within the region. However, assuming that the dust constitution is homogeneous biases the derived properties. The most problematic aspect is certainly the variation of the overall FIR opacity. If there is significant mantle accretion in dense regions, we might be mixing together regions with different κ (cf. Fig. I.21). This is unavoidable and this has to be pondered when discussing the modeling results.

**The degeneracy** between the grain size and ISRF distributions prevents constraining the former. This is because a MIR excess due to an enhancement of the small grain emission looks similar to the presence of hot equilibrium grains (such as compact H II regions) Fig. III.15 demonstrates this degeneracy. It shows the fit of the same synthetic observations, either varying the fraction of small grains, or the mixing of ISRFs. Additional constraints, on the geometry of the region and its radiation field, are necessary to attempt breaking this degeneracy (cf. Sect. III.1.3.4).

**An alternative distribution.** The starlight distribution of Eq. (III.36) is not the only possible one. Draine & Li (2007) proposed the following:

$$ \left( \frac{1}{M_{\text{dust}}} \frac{dM_{\text{dust}}}{dU} \right)_{\text{DL07}} = \gamma \times \left( \frac{1}{M_{\text{dust}}} \frac{dM_{\text{dust}}}{dU} \right)_{\text{Eq. (III.36)}} + (1 - \gamma) \times \delta(U - U_{-}), \quad (\text{III.40}) $$

<figure><img src="image_298.png" /></figure>
<figure><img src="image_299.png" /></figure>

<figure><img src="image_300.png" /></figure>

FIGURE III.15 – Degeneracy of the grain size and ISRF distributions. The two panels show the same synthetic observations (black error bars) fitted by two different models (Galliano et al., 2018). Panel (a) represents a fit by playing on the proportions of grains of different sizes: PAHs, VSGs and BGs. Panel (b) represents a fit using Eq. (III.36). In both cases, we have used the THEMIS dust mixture. Licensed under CC BY-SA 4.0.

which simply is the linear combination of the power-law distribution of Eq. (III.36), with a Dirac distribution centered at $U = U_-$, fixing $\alpha = 2$ and $U_- + \Delta U = 10^6$. The power-law distribution is supposed to account for star-forming regions, with a large gradient of starlight intensity, and the Dirac represents the diffuse ISM, uniformly illuminated. The parameter $\gamma$ controls the relative weight of these two components. This distribution was elaborated when Spitzer data were being analyzed. There was no coverage beyond $\lambda = 160$ µm. There was thus no constraint on the Rayleigh-Jeans regime of the SED. The role of the Dirac component was to fit the SED peak, avoiding the dust mass to explode by lack of constraint on the cold dust distribution. This parametrization however became problematic when Herschel data arrived. The FIR-submm slope of the observed SED can indeed not be fitted as well with this model (Eq. III.40) as with the composite approach (Eq. III.36). This is because the model's slope is the intrinsic slope of the grain mixture. This is demonstrated in Fig. III.16.a. In Fig. III.16.b, we see that the starlight intensity distribution fit can not go down as low as the composite model. The Dirac component fits the FIR peak with a compromise $U_-$. This is reminiscent of the discussion we had about MBB fits, in Sect. III.1.2.1. This was demonstrated by G21, who compared different approaches by fitting the SEDs of about 800 galaxies of the DustPedia project (Davies et al., 2017) and Dwarf Galaxy Sample (DGS; Madden et al., 2013). Fig. III.17 compares the composite approach, as a reference, to the following models (see the complete discussion in G21).

**A MBB with $\beta$ free** infers dust masses that are, on average over the whole sample, consistent with the composite approach (Fig. III.17.a). However, we see there is a bias as a function of the mass of the galaxy. The MBB approach tends to overestimate the mass of late-type galaxies (mostly the high-mass objects), and underestimates the mass of early-type galaxies. This is because the SED of early-type galaxies is fitted with a smaller $\beta$ (thus a lower temperature, and a lower dust mass).

**A MBB with $\beta$ fixed** tends to underestimate the dust mass on average. In Fig. III.17.b, the median of the ratio is $\approx 0.85$. This is the effect we have discussed in Sect. III.1.2.1: the mixing of physical conditions is fitted by a compromise temperature, underestimating the coldest dust.

<figure><img src="image_301.png" /></figure>
<figure><img src="image_302.png" /></figure>

**The Draine & Li (2007) distribution (Eq. III.40)** results in a similar bias as the MBB. For the distribution in Fig. III.17.c, the median of the ratio is ≈ 0.71, even lower than in the case of the MBB. The reason is the same: the uniformly illuminated component dominating the FIR-submm SED can not account for the distribution of cold dust.

<figure><img src="image_303.png" /></figure>

FIGURE III.16 – SED fit with the Draine & Li (2007) starlight intensity distribution. The black error bars in panel (a) are synthetic observations. They are fitted using Eq. (III.40): (i) the power-law component in magenta; (ii) the Dirac component in blue; and (iii) an additional stellar continuum in yellow. The corresponding starlight intensity distribution is shown in panel (b), with the same color code. We have overlaid, in green, a composite model fit to the same synthetic observations. Licensed under CC BY-SA 4.0.

### III.1.2.3 Panchromatic Empirical SED Models

Several codes in the literature model the *panchromatic*<fnref n="3" /> SED of galaxies, with a simplified treatment of the radiative transfer (e.g. Silva et al., 1998; Charlot & Fall, 2000; Galliano et al., 2008a; da Cunha et al., 2008; Boquien et al., 2019; Fioc & Rocca-Volmerange, 2019). They include the emission by stellar populations, in addition to dust SEDs.

**Stellar SEDs.** Stars of different masses have different spectra and lifetimes. This is shown in Fig. III.18, in the form of a *Hertzsprung-Russell diagram*. Massive stars have: (i) the highest effective temperatures, their SEDs peaking in the UV; (ii) the highest luminosities; and (iii) the shortest lifetimes, of only a few million years. This is the opposite for low-mass stars: their SED peaks in the NIR, and they live longer than several hundred million years. These characteristics have a profound impact on the variation of stellar SEDs with time. The intrinsic emission of a stellar population can be simulated using *evolutionary synthesis* (e.g. Fioc & Rocca-Volmerange, 1997). This approach follows, at each time step, the formation and evolution of stars with different masses, $m_★$. When these stars are born, all populations contribute to the SED. It is shown as the magenta curve in Fig. III.19. This initial SED is dominated by massive stars and peaks in the UV. Then, as the most massive stars, which also have the shortest lifetime, die, their contribution to the SED is suppressed. Consequently, when these stars disappear, the UV-side of the SED decreases. After several hundred million years, the SED

<figure><img src="image_304.png" /></figure>

<footnote n="3">3. Literally for *all wavelengths*, from the X-rays to the radio.</footnote>
<figure><img src="image_305.png" /></figure>

<figure><img src="image_306.png" /><figcaption>FIGURE III.17 – *Comparison of different SED models.* The data are from the SED modeling of the DustPedia and DGS nearby galaxies by Galliano et al. (2021, ≈ 800 galaxies of all types). The x-axis of the left panels shows the dust mass derived by fitting the SED of each galaxy using the composite approach (Eq. III.36). The y-axis shows the ratio of the x-axis and the dust mass estimate using one of the three alternate models: (i) MBB; (ii) MBB with fixed β = 1.79; (iii) Draine & Li (2007) distribution. Each point with an error bar corresponds to one particular galaxy. The yellow line shows the 1:1 ratio, and the dashed red line, the median of the sample. The right panels show the histogram of the distribution. Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_307.png" /></figure>

peaks in the NIR, as it originates only in low-mass stars (orange and red curves in Fig. III.19). There is also a drastic evolution of the emissivity as a function of time, as low-mass stars are significantly less luminous (cf. Fig. III.18). To compute such synthetic spectra, the following quantities need to be defined.

* The *Star Formation History (SFH)* expresses the variation as a function of time of the *Star Formation Rate (SFR)*, $\psi(t) = dm_{\star}/dt$. Several forms are used in the literature: instantaneous burst, exponential, delayed, etc. (e.g. Lee et al., 2010).

* The *Initial Mass Function (IMF)* gives the *PDF* to form a star of mass $m_{\star}$. The historical IMF of Salpeter (1955) is less commonly used nowadays. For instance, an alternate IMF has been proposed by Chabrier (2003).

We will more extensively discuss these quantities in Sect. IV.3, when modeling the chemical evolution of galaxies. We have summarized in Table III.2 the main properties of the different stellar classes.

<table><caption>TABLE III.2 – Basic properties of the main stellar classes.</caption><thead><tr><th></th><th>Effective temperature, T<sub>eff</sub></th><th>Initial mass, m<sub>★</sub></th><th>Initial luminosity, L<sub>★</sub></th><th>Lifetime, τ(m<sub>★</sub>)</th></tr></thead><tbody><tr><td>O</td><td>≥ 30 000 K</td><td>≥ 16 M<sub>☉</sub></td><td>≥ 30 000 L<sub>☉</sub></td><td>≤ 12 Myr</td></tr><tr><td>B</td><td>10 000 – 30 000 K</td><td>2.1 – 16 M<sub>☉</sub></td><td>25 – 30 000 L<sub>☉</sub></td><td>12 Myr – 1 Gyr</td></tr><tr><td>A</td><td>7 500 – 10 000 K</td><td>1.4 – 2.1 M<sub>☉</sub></td><td>5 – 25 L<sub>☉</sub></td><td>1 – 3 Gyr</td></tr><tr><td>F</td><td>6 000 – 7 500 K</td><td>1.04 – 1.4 M<sub>☉</sub></td><td>1.5 – 5 L<sub>☉</sub></td><td>3 – 10 Gyr</td></tr><tr><td>G</td><td>5 200 – 6 000 K</td><td>0.8 – 1.04 M<sub>☉</sub></td><td>0.6 – 1.5 L<sub>☉</sub></td><td>10 – 25 Gyr</td></tr><tr><td>K</td><td>3 700 – 5 200 K</td><td>0.45 – 0.8 M<sub>☉</sub></td><td>0.08 – 0.6 L<sub>☉</sub></td><td>...</td></tr><tr><td>M</td><td>2 400 – 3 700 K</td><td>0.08 – 0.45 M<sub>☉</sub></td><td>≤ 0.08 L<sub>☉</sub></td><td>...</td></tr></tbody></table>

**Putting everything together.** Empirical panchromatic SED models usually include the following physical ingredients.

**One or several stellar populations** are modeled. Their escaping radiation is fitted to the UV-to-NIR SED, allowing the user to constrain: the age, the SFH timescale, the SFR and the total stellar mass. The *attenuation*<fnref n="4" /> is accounted for, by assuming simple topologies, such as those discussed in Calzetti et al. (1994). The intrinsic stellar power, $L_{\star}$, is derived from the *energy balance*: $L_{\star} = L_{UV-NIR} + L_{TIR}$, where $L_{UV-NIR}$ is the escaping UV-to-NIR power. This equation simply states that the intrinsic power emitted by stars is either escaping the galaxy in the UV-to-NIR range, or has been absorbed by dust. It implicitly assumes that the emission from the galaxy is isotropic, which is not necessarily the case for disk galaxies.

**Dust emission** is modeled, either by solving the radiative transfer equation (Eq. III.16) in a simple geometry, or by adopting a distribution of starlight intensities (cf. Sect. III.1.2.2). The dust properties are not always consistent with those used to account for attenuation.

**Nebular emission**, in the form of lines or continuum, can be included. In case of star-forming galaxies, it is dominated by HII regions (e.g. Charlot & Longhetti, 2001). Some models also include the gaseous emission from AGNs. These tracers are used to refine some of the diagnostics.

We illustrate this approach with the model of Galliano et al. (2008a), applied to two galaxies, in Fig. III.20. This particular model separates the emission from HII regions, which are powered by massive ionizing stars (cyan). The magenta curve shows the escaping radiation from HII regions. It includes

<figure><img src="image_308.png" /></figure>

<footnote n="4">4. Attenuation is not equivalent to extinction. The extinction is the sum of scattering and absorption along the sightline toward a point source, whereas the attenuation is a global account of the reddening of an ensemble of stars, potentially mixed with the dust. The extinction is directly linked to dust properties and is independent of geometry. The attenuation is a synthetic quantity depending both on the dust properties and on the ISM topology (e.g. Buat et al., 2019, for a recent discussion).</footnote>
<figure><img src="image_309.png" /><figcaption>FIGURE III.18 – *Stellar isochrones*. This figure shows theoretical tracks of individual star evolution from the model of Schaller et al. (1992), at initial metallicity, Z = 0.008. This representation is known as a *Hertzsprung-Russell diagram*. Stars of a given mass, $m_★$ (a given color), start from the *Zero Age Main Sequence* (ZAMS; grey) and evolve along their *Main Sequence* (MS) track until their death, after a time $τ(m_★)$. We have highlighted the late evolutionary stages: (i) red supergiants for massive stars; and (ii) AGB for *Low- and Intermediate-Mass Stars* (LIMS). Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_310.png" /></figure>

<figure><img src="image_311.png" /><figcaption>FIGURE III.19 – *Evolution of stellar SEDs as a function of time.* The different curves show the evolution, as a function of time, of the SED of a stellar population produced by an instantaneous burst, at Solar metallicity. We have assumed the Salpeter (1955) IMF. These models were generated using the code of Fioc & Rocca-Volmerange (1997). Notice, in particular, the fast decrease of ionizing photons with time. Licensed under CC BY-SA 4.0.</figcaption></figure>

the dust emission and the free-free continuum. The escaping radiation from H II regions, as well as non-ionizing field stars (yellow), heat the neutral ISM (red). The degeneracy between both dust components was solved by using radio observations to constrain the level of the free-free emission. The gas density in H II regions, impacting the equilibrium temperature of large grains, is constrained by the MIR continuum. The remaining emission is thus assumed to come from the neutral ISM. We will discuss in more detail the results of this model, in Sect. IV.3.

**Limitations of empirical panchromatic models.** The approach we have just described has several advantages. In a single fit, it provides estimates of the SFR, the age of the stellar populations, the stellar mass, and the dust properties. Its major limitation however resides in the sensitivity of its results to the assumed ISM topology. The ISM indeed has a fractal structure, with several orders of magnitude of contrast density (e.g. Combes, 2000). The optical depth from the model thus probably underestimates the total dust column density (cf. Sect. III.1.1.3). In addition, the extinction and emission properties of these models are usually not consistent. The modeling of the microscopic grain constitution and of their macroscopic spatial distribution can differ from one side of the electromagnetic spectrum to the other. For instance, assuming a Calzetti et al. (1994) attenuation law and taking a mean radiation field to account for dust heating is virtually equivalent to decoupling the extinction and emission. Finally, the consistency brought by modeling all multi-wavelength tracers at once, which is a priori positive, leads to propagating the systematic uncertainties, due to arbitrary choices of ISM topologies, to parameters that could have been considered independent of these effects, if they had been modeled separately ($M_{dust}$, $\langle U \rangle$, $q_{PAH}$; Sect. III.1.2.2).

<figure><img src="image_312.png" /></figure>
<figure><img src="image_313.png" /></figure>

<figure><img src="image_314.png" /><figcaption>FIGURE III.20 – Multiphase SEDs of galaxies. This figure shows the SED of two galaxies, modeled by Galliano et al. (2008a): the prototypical starburst, M 82, and the lowest metallicity galaxy, I Zw 18. The black error bars represent the observed fluxes. The total SED model is shown in grey. The stars are split in two populations: (i) young, ionizing stars that are in HII regions; and (ii) older, non-ionizing stars that are in the rest of the galaxy. We show the intrinsic emission of both stellar components. The dust emission originates in: (i) HII regions, heated by ionizing stars; and (ii) the neutral ISM, heated by non-ionizing stars and by the escaping radiation from HII regions. Licensed under CC BY-SA 4.0.</figcaption></figure>