# A Nearby Galaxy Perspective on Interstellar Dust Properties and their Evolution

Frédéric Galliano

► To cite this version:

Frédéric Galliano. A Nearby Galaxy Perspective on Interstellar Dust Properties and their Evolution. Cosmology and Extra-Galactic Astrophysics [astro-ph.CO]. Université Paris-Saclay, 2022. tel-03555658

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Submitted on 3 Feb 2022

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<figure><img src="image_1.png" /></figure>

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# A Nearby Galaxy Perspective on Interstellar Dust Properties and their Evolution

Habilitation à diriger des recherches de l'Université Paris-Saclay

Habilitation présentée et soutenue à Gif-sur-Yvette, le vendredi 14 janvier 2022, par

Frédéric GALLIANO

Département d'Astrophysique, CEA Paris-Saclay

## Composition du jury:

**Véronique BUAT**
Professeure, Laboratoire d'Astrophysique de Marseille
Rapportrice

**Stéphane CHARLOT**
Directeur de recherche, Institut d'Astrophysique de Paris
Examineur

**Vassilis CHARMANDARIS**
Professeur, Université de Crète, Grèce
Rapporteur

**François-Xavier DÉSERT**
Astronome, Institut de Planétologie et d'Astrophysique de Grenoble
Examineur

**Thomas HENNING**
Professeur, Institut Max Planck d'Astronomie, Heidelberg, Allemagne
Rapporteur

**Laurent VERSTRAETE**
Professeur, Institut d'Astrophysique Spatiale, Orsay
Président
<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /></figure>

# Contents

Contents iii

List of Figures ix

List of Tables xiv

Abstract xvii

Introduction xix

Interstellar Dust: A Key To Understanding Galaxy Evolution xix

The Relevance of Nearby Galaxies xx

Scope of the Manuscript xxi

I Propaedeutics in Dust Physics 1

I.1 The Make-Up of Solids 3

I.1.1 Atomic Structure 3

I.1.1.1 The Hydrogen Atom 3

I.1.1.2 Polyelectronic Atoms 3

I.1.1.3 The Valence Shell 4

I.1.1.4 Orbital Hybridisation 7

I.1.2 Molecular Bonding 7

I.1.2.1 Covalent Bonds 7

I.1.2.2 Ionic Bonds 9

I.1.2.3 Metallic Bonds 10

I.1.2.4 Intermolecular Attraction 10

I.1.3 The Solid State 10

I.1.3.1 The Different Types of Solids 10

I.1.3.2 The Band Structure of Solids 10

I.1.3.3 The Fermi Level 11

I.1.4 Interstellar Dust Candidates 12

I.1.4.1 Silicates 12

I.1.4.2 Hydrogenated Amorphous Carbon 13

I.1.4.3 Graphite 15

I.1.4.4 Polycyclic Aromatic Hydrocarbons (PAH) 15

I.2 The Interaction of Light with Solids 15

I.2.1 Bonds as Harmonic Oscillators 17

I.2.1.1 The Harmonic Oscillator Amplitude 17

I.2.1.2 The Plasma Frequency 17

I.2.1.3 The Dielectric Function 18

I.2.1.4 Harmonic Oscillator Cross-Section 19

I.2.1.5 Optical Constants of Conductors 21

I.2.1.6 The Kramers-Kronig Relations 22

I.2.2 Grain Optical Properties 23

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /></figure>

<table>
  <tbody>
    <tr>
      <td>I.2.2.1</td>
      <td>Why Are Most Dust Features in the MIR?</td>
      <td>23</td>
    </tr>
    <tr>
      <td>I.2.2.2</td>
      <td>Dielectric Functions of Realistic Materials</td>
      <td>23</td>
    </tr>
    <tr>
      <td>I.2.2.3</td>
      <td>Computing Grain Cross-Sections</td>
      <td>24</td>
    </tr>
    <tr>
      <td>I.2.2.4</td>
      <td>Beyond Homogeneous Spheres</td>
      <td>30</td>
    </tr>
    <tr>
      <td>I.2.2.5</td>
      <td>Polarization</td>
      <td>32</td>
    </tr>
    <tr>
      <td>I.2.3</td>
      <td>Heat Capacities</td>
      <td>34</td>
    </tr>
    <tr>
      <td>I.2.3.1</td>
      <td>Distribution of Harmonic Oscillators</td>
      <td>34</td>
    </tr>
    <tr>
      <td>I.2.3.2</td>
      <td>Debye's Model</td>
      <td>36</td>
    </tr>
    <tr>
      <td>I.2.3.3</td>
      <td>Heat Capacities of Realistic Materials</td>
      <td>37</td>
    </tr>
    <tr>
      <td>I.2.4</td>
      <td>Heating and Cooling</td>
      <td>37</td>
    </tr>
    <tr>
      <td>I.2.4.1</td>
      <td>Kirchhoff's Law</td>
      <td>37</td>
    </tr>
    <tr>
      <td>I.2.4.2</td>
      <td>Equilibrium Heating</td>
      <td>40</td>
    </tr>
    <tr>
      <td>I.2.4.3</td>
      <td>Stochastic Heating</td>
      <td>42</td>
    </tr>
    <tr>
      <td>I.2.4.4</td>
      <td>Collisionnal Heating</td>
      <td>48</td>
    </tr>
    <tr>
      <td><strong>II</strong></td>
      <td><strong>Dust Observables and Models</strong></td>
      <td><strong>51</strong></td>
    </tr>
    <tr>
      <td>II.1</td>
      <td>A Brief History of Interstellar Dust Studies</td>
      <td>53</td>
    </tr>
    <tr>
      <td>II.1.1</td>
      <td>The Challenges of Observing Interstellar Regions</td>
      <td>55</td>
    </tr>
    <tr>
      <td>II.1.1.1</td>
      <td>Limitations Due to the Atmosphere</td>
      <td>55</td>
    </tr>
    <tr>
      <td>II.1.1.2</td>
      <td>Historical Ground-Based Observatories</td>
      <td>55</td>
    </tr>
    <tr>
      <td>II.1.1.3</td>
      <td>Airborne Observatories</td>
      <td>56</td>
    </tr>
    <tr>
      <td>II.1.1.4</td>
      <td>Space Telescopes</td>
      <td>57</td>
    </tr>
    <tr>
      <td>II.1.1.5</td>
      <td>Grain-Collecting Spacecrafts</td>
      <td>61</td>
    </tr>
    <tr>
      <td>II.1.2</td>
      <td>Chronology of the Main Breakthroughs</td>
      <td>62</td>
    </tr>
    <tr>
      <td>II.1.2.1</td>
      <td>Obscuration and Dimming of Starlight</td>
      <td>62</td>
    </tr>
    <tr>
      <td>II.1.2.2</td>
      <td>The Dust Continuum</td>
      <td>63</td>
    </tr>
    <tr>
      <td>II.1.2.3</td>
      <td>Identification of Dust Features</td>
      <td>64</td>
    </tr>
    <tr>
      <td>II.1.2.4</td>
      <td>Dusty Epiphenomena</td>
      <td>66</td>
    </tr>
    <tr>
      <td>II.1.2.5</td>
      <td>Dust Models</td>
      <td>66</td>
    </tr>
    <tr>
      <td>II.2</td>
      <td>The Current Empirical Constraints</td>
      <td>68</td>
    </tr>
    <tr>
      <td>II.2.1</td>
      <td>Extinction</td>
      <td>68</td>
    </tr>
    <tr>
      <td>II.2.1.1</td>
      <td>UV-to-NIR Extinction</td>
      <td>70</td>
    </tr>
    <tr>
      <td>II.2.1.2</td>
      <td>MIR Extinction</td>
      <td>72</td>
    </tr>
    <tr>
      <td>II.2.1.3</td>
      <td>X-Rays</td>
      <td>73</td>
    </tr>
    <tr>
      <td>II.2.1.4</td>
      <td>Dichroic Extinction</td>
      <td>75</td>
    </tr>
    <tr>
      <td>II.2.1.5</td>
      <td>Diffuse Interstellar Bands</td>
      <td>75</td>
    </tr>
    <tr>
      <td>II.2.2</td>
      <td>Emission</td>
      <td>76</td>
    </tr>
    <tr>
      <td>II.2.2.1</td>
      <td>Infrared Continuum and Features</td>
      <td>76</td>
    </tr>
    <tr>
      <td>II.2.2.2</td>
      <td>Polarized Emission</td>
      <td>77</td>
    </tr>
    <tr>
      <td>II.2.2.3</td>
      <td>Non-Thermal Emission</td>
      <td>77</td>
    </tr>
    <tr>
      <td>II.2.3</td>
      <td>Elemental abundances in Grains</td>
      <td>78</td>
    </tr>
    <tr>
      <td>II.2.3.1</td>
      <td>Measuring ISM abundances</td>
      <td>78</td>
    </tr>
    <tr>
      <td>II.2.3.2</td>
      <td>Depletions</td>
      <td>79</td>
    </tr>
    <tr>
      <td>II.2.4</td>
      <td>Direct Measures</td>
      <td>81</td>
    </tr>
    <tr>
      <td>II.2.4.1</td>
      <td>Meteorite Inclusions</td>
      <td>82</td>
    </tr>
    <tr>
      <td>II.2.4.2</td>
      <td>Interplanetary Dust</td>
      <td>83</td>
    </tr>
    <tr>
      <td>II.2.4.3</td>
      <td>Laboratory Measurements</td>
      <td>83</td>
    </tr>
    <tr>
      <td>II.3</td>
      <td>State-of-the-Art Dust Models</td>
      <td>85</td>
    </tr>
    <tr>
      <td>II.3.1</td>
      <td>Composition and Size Distributions of Different Models</td>
      <td>85</td>
    </tr>
    <tr>
      <td>II.3.1.1</td>
      <td>Diversity in Composition</td>
      <td>85</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /></figure>

<table>
  <tbody>
    <tr>
      <td>II.3.1.2 Difference in Size Distributions</td>
      <td>86</td>
    </tr>
    <tr>
      <td>II.3.2 The Model Properties</td>
      <td>87</td>
    </tr>
    <tr>
      <td>II.3.2.1 Extinction and Emission</td>
      <td>88</td>
    </tr>
    <tr>
      <td>II.3.2.2 The Fitted Constraints</td>
      <td>89</td>
    </tr>
    <tr>
      <td>II.3.3 Some Useful Quantities</td>
      <td>89</td>
    </tr>
    <tr>
      <td>II.3.3.1 Grain Sizes, Areas and Masses</td>
      <td>89</td>
    </tr>
    <tr>
      <td>II.3.3.2 Opacity and Emissivity</td>
      <td>93</td>
    </tr>
    <tr>
      <td><strong>III The Grain Properties of Nearby Galaxies</strong></td>
      <td><strong>97</strong></td>
    </tr>
    <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>III.1.1.2 The Radiative Transfer Equation</td>
      <td>103</td>
    </tr>
    <tr>
      <td>III.1.1.3 Approximations for Clumpy Media</td>
      <td>108</td>
    </tr>
    <tr>
      <td>III.1.1.4 Rigorous Solutions</td>
      <td>109</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>III.1.2.3 Panchromatic Empirical SED Models</td>
      <td>122</td>
    </tr>
    <tr>
      <td>III.1.2.4 The Matryoshka Effect</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>III.1.3.2 Large-Scale Dust Distribution in Galaxies</td>
      <td>131</td>
    </tr>
    <tr>
      <td>III.1.3.3 Constraining the Grain Opacity</td>
      <td>135</td>
    </tr>
    <tr>
      <td>III.1.3.4 Constraining the Size Distribution</td>
      <td>138</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>III.2.1.2 Spectral Decomposition Methods</td>
      <td>144</td>
    </tr>
    <tr>
      <td>III.2.1.3 PAH Band Ratio Studies</td>
      <td>151</td>
    </tr>
    <tr>
      <td>III.2.1.4 Variations of the Aromatic Feature Strength</td>
      <td>157</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>III.2.2.2 The Anomalous Microwave Emission</td>
      <td>162</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>III.3.1.1 The Neutral Atomic Gas</td>
      <td>166</td>
    </tr>
    <tr>
      <td>III.3.1.2 The Ionized Gas</td>
      <td>168</td>
    </tr>
    <tr>
      <td>III.3.1.3 The Molecular Gas</td>
      <td>170</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>III.3.2.2 Photodissociation Regions</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><strong>IV Modeling Cosmic Dust Evolution</strong></td>
      <td><strong>177</strong></td>
    </tr>
    <tr>
      <td>IV.1 Stellar Evolution</td>
      <td>180</td>
    </tr>
    <tr>
      <td>IV.1.1 The Fate of Stars of Different Masses</td>
      <td>180</td>
    </tr>
    <tr>
      <td>IV.1.1.1 Nucleosynthesis</td>
      <td>180</td>
    </tr>
    <tr>
      <td>IV.1.1.2 Brief Outline of Stellar Evolution</td>
      <td>181</td>
    </tr>
    <tr>
      <td>IV.1.1.3 Parametrizing Star Formation</td>
      <td>183</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /></figure>

<table><tbody><tr><td>IV.1.2</td><td>Elemental and Dust Yields</td><td>185</td></tr><tr><td>IV.1.2.1</td><td>Injection of Heavy Elements in the ISM</td><td>186</td></tr><tr><td>IV.1.2.2</td><td>Production of Stardust</td><td>188</td></tr><tr><td>IV.2</td><td>Dust Evolution Processes in the ISM</td><td>189</td></tr><tr><td>IV.2.1</td><td>Grain Formation and Transformation</td><td>189</td></tr><tr><td>IV.2.1.1</td><td>Evidence of Grain Growth and Coagulation in the ISM</td><td>190</td></tr><tr><td>IV.2.1.2</td><td>Studies of the Magellanic Clouds</td><td>190</td></tr><tr><td>IV.2.1.3</td><td>Quantifying Grain Growth</td><td>192</td></tr><tr><td>IV.2.2</td><td>Grain Destruction</td><td>195</td></tr><tr><td>IV.2.2.1</td><td>Photodestruction of Small Grains</td><td>195</td></tr><tr><td>IV.2.2.2</td><td>Thermal Sputtering</td><td>196</td></tr><tr><td>IV.2.2.3</td><td>Destruction by SN Blast Waves</td><td>199</td></tr><tr><td>IV.3</td><td>Cosmic Dust Evolution</td><td>201</td></tr><tr><td>IV.3.1</td><td>Constraining the Dust Build-Up in Galaxies</td><td>201</td></tr><tr><td>IV.3.1.1</td><td>Cosmic Dust Evolution Models</td><td>202</td></tr><tr><td>IV.3.1.2</td><td>Empirical Inference of Dust Evolution Timescales</td><td>206</td></tr><tr><td>IV.3.2</td><td>Evolution of the Aromatic Feature Carriers</td><td>212</td></tr><tr><td>IV.3.2.1</td><td>The Different Evolution Scenarios</td><td>213</td></tr><tr><td>IV.3.2.2</td><td>The Observed Trends</td><td>214</td></tr></tbody></table>

V Methodological Effort and Epistemological Reflection 217

<table><tbody><tr><td>V.1</td><td>Understanding the Opposition between Bayesians and Frequentists</td><td>219</td></tr><tr><td>V.1.1</td><td>Two Conceptions of Probability and Uncertainty</td><td>220</td></tr><tr><td>V.1.1.1</td><td>The Concept of Conditional Probability</td><td>220</td></tr><tr><td>V.1.1.2</td><td>The Bayesian and Frequentist Assumptions</td><td>221</td></tr><tr><td>V.1.2</td><td>Comparison of the Two Approaches on Simple Cases</td><td>223</td></tr><tr><td>V.1.2.1</td><td>Simple Case: When the Two Approaches Agree</td><td>223</td></tr><tr><td>V.1.2.2</td><td>Benefits of Using an Informative Prior</td><td>225</td></tr><tr><td>V.1.2.3</td><td>Case Where the Two Approaches Differ: Non Gaussianity and Few Data</td><td>227</td></tr><tr><td>V.1.3</td><td>Numerical Methods to Solve Bayesian Problems</td><td>229</td></tr><tr><td>V.1.3.1</td><td>Sampling the Posterior Distribution</td><td>229</td></tr><tr><td>V.1.3.2</td><td>Post-Processing MCMCs</td><td>232</td></tr><tr><td>V.1.4</td><td>Decision Making and Limitations of the Frequentist Approach</td><td>238</td></tr><tr><td>V.1.4.1</td><td>Hypothesis Testing</td><td>238</td></tr><tr><td>V.1.4.2</td><td>Pros and Cons of the two Approaches</td><td>241</td></tr><tr><td>V.2</td><td>Bayesianism, an Alternative to Popper's Scientific Method</td><td>243</td></tr><tr><td>V.2.1</td><td>Bayes Formula Throughout History</td><td>243</td></tr><tr><td>V.2.1.1</td><td>The Origins</td><td>243</td></tr><tr><td>V.2.1.2</td><td>The Frequentist Winter</td><td>245</td></tr><tr><td>V.2.1.3</td><td>The Bayesian Renaissance</td><td>246</td></tr><tr><td>V.2.2</td><td>Bayesian and Popperian Epistemologies</td><td>248</td></tr><tr><td>V.2.2.1</td><td>The Epistemological Debate at the Beginning of the XXth Century</td><td>248</td></tr><tr><td>V.2.2.2</td><td>Popper's Logic of Scientific Discovery</td><td>249</td></tr><tr><td>V.2.2.3</td><td>Verifiability, Holisticity and Parsimony: the Bayesian Alternative</td><td>250</td></tr><tr><td>V.3</td><td>Relevance for Interstellar Dust Studies</td><td>253</td></tr><tr><td>V.3.1</td><td>The Particularities of Interstellar Dust Studies</td><td>254</td></tr><tr><td>V.3.1.1</td><td>Complexity of the Physics</td><td>254</td></tr></tbody></table>
<figure><img src="image_10.png" /></figure>

<table>
  <tbody>
    <tr>
      <td>V.3.1.2</td>
      <td>Entanglement of the Observations</td>
      <td>254</td>
    </tr>
    <tr>
      <td>V.3.2</td>
      <td>The Principles of Hierarchical Bayesian Inference</td>
      <td>255</td>
    </tr>
    <tr>
      <td>V.3.2.1</td>
      <td>Non-Hierarchical Bayesian Formalism for SED Modeling</td>
      <td>255</td>
    </tr>
    <tr>
      <td>V.3.2.2</td>
      <td>The Introduction of Nuisance Variables</td>
      <td>256</td>
    </tr>
    <tr>
      <td>V.3.2.3</td>
      <td>The Role of the Hyperparameters</td>
      <td>257</td>
    </tr>
    <tr>
      <td>V.3.3</td>
      <td>Hierarchical Bayesian Models for ISD Studies</td>
      <td>258</td>
    </tr>
    <tr>
      <td>V.3.3.1</td>
      <td>Efficiency and Comparison to Other Methods</td>
      <td>258</td>
    </tr>
    <tr>
      <td>V.3.3.2</td>
      <td>The Role of the Prior</td>
      <td>262</td>
    </tr>
    <tr>
      <td>V.3.3.3</td>
      <td>Other Developments</td>
      <td>264</td>
    </tr>
    <tr>
      <td><strong>VI Conclusion and Prospective</strong></td>
      <td></td>
      <td><strong>267</strong></td>
    </tr>
    <tr>
      <td>VI.1</td>
      <td>What Have We Learned About ISD in the Past Decade?</td>
      <td>268</td>
    </tr>
    <tr>
      <td>VI.1.1</td>
      <td>About Dust Properties</td>
      <td>268</td>
    </tr>
    <tr>
      <td>VI.1.2</td>
      <td>About Dust Evolution</td>
      <td>269</td>
    </tr>
    <tr>
      <td>VI.2</td>
      <td>What Are the Open Questions for the Next Decade?</td>
      <td>270</td>
    </tr>
    <tr>
      <td>VI.2.1</td>
      <td>Extragalactic Dust</td>
      <td>270</td>
    </tr>
    <tr>
      <td>VI.2.2</td>
      <td>Dust Evolution Modeling</td>
      <td>271</td>
    </tr>
    <tr>
      <td>VI.2.3</td>
      <td>Dusty Epiphenomena</td>
      <td>271</td>
    </tr>
    <tr>
      <td>VI.2.4</td>
      <td>The Need for a New FIR Observatory</td>
      <td>272</td>
    </tr>
    <tr>
      <td>VI.2.5</td>
      <td>The Public Image of Interstellar Dust</td>
      <td>272</td>
    </tr>
    <tr>
      <td>VI.3</td>
      <td>Current Future Projects</td>
      <td>272</td>
    </tr>
    <tr>
      <td>VI.3.1</td>
      <td>The Modelosaur Approach</td>
      <td>272</td>
    </tr>
    <tr>
      <td>VI.3.2</td>
      <td>Out-of-the-Box Idea Bin</td>
      <td>273</td>
    </tr>
    <tr>
      <td><strong>A List of Acronyms</strong></td>
      <td></td>
      <td><strong>275</strong></td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>General Acronyms</td>
      <td>275</td>
    </tr>
    <tr>
      <td>A.2</td>
      <td>Telescope and Instrument Acronyms</td>
      <td>277</td>
    </tr>
    <tr>
      <td>A.3</td>
      <td>Model and Project Acronyms</td>
      <td>278</td>
    </tr>
    <tr>
      <td>A.4</td>
      <td>Denomination of the Main Spectral Windows</td>
      <td>279</td>
    </tr>
    <tr>
      <td><strong>B Astronomers and Units</strong></td>
      <td></td>
      <td><strong>281</strong></td>
    </tr>
    <tr>
      <td>B.1</td>
      <td>Brief History of Unit Systems</td>
      <td>281</td>
    </tr>
    <tr>
      <td>B.2</td>
      <td>Working with Units</td>
      <td>282</td>
    </tr>
    <tr>
      <td><strong>C Useful Formulae</strong></td>
      <td></td>
      <td><strong>289</strong></td>
    </tr>
    <tr>
      <td>C.1</td>
      <td>3D Quantities and Volume Integrals</td>
      <td>290</td>
    </tr>
    <tr>
      <td>C.1.1</td>
      <td>Differential Operators</td>
      <td>290</td>
    </tr>
    <tr>
      <td>C.1.1.1</td>
      <td>Gradient</td>
      <td>290</td>
    </tr>
    <tr>
      <td>C.1.1.2</td>
      <td>Laplacian</td>
      <td>291</td>
    </tr>
    <tr>
      <td>C.1.1.3</td>
      <td>Divergence</td>
      <td>291</td>
    </tr>
    <tr>
      <td>C.1.1.4</td>
      <td>Curl</td>
      <td>291</td>
    </tr>
    <tr>
      <td>C.1.2</td>
      <td>Vectorial Analysis</td>
      <td>291</td>
    </tr>
    <tr>
      <td>C.1.3</td>
      <td>Integral Theorems</td>
      <td>292</td>
    </tr>
    <tr>
      <td>C.1.4</td>
      <td>Dust Heating and Cooling: Two Ways of Slicing the Pis</td>
      <td>292</td>
    </tr>
    <tr>
      <td>C.1.4.1</td>
      <td>Solution 1</td>
      <td>293</td>
    </tr>
    <tr>
      <td>C.1.4.2</td>
      <td>Solution 2</td>
      <td>293</td>
    </tr>
    <tr>
      <td>C.2</td>
      <td>Statistics</td>
      <td>293</td>
    </tr>
    <tr>
      <td>C.2.1</td>
      <td>General Formulae</td>
      <td>293</td>
    </tr>
    <tr>
      <td>C.2.1.1</td>
      <td>Moments of a PDF</td>
      <td>293</td>
    </tr>
    <tr>
      <td>C.2.1.2</td>
      <td>Marginalization</td>
      <td>294</td>
    </tr>
    <tr>
      <td>C.2.1.3</td>
      <td>Variable Change</td>
      <td>294</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /></figure>

<table><tbody><tr><td>C.2.1.4 Combining Uncertainties</td><td>294</td></tr><tr><td>C.2.2 Useful Probability Distributions</td><td>295</td></tr><tr><td>C.2.2.1 Binomial Distribution</td><td>295</td></tr><tr><td>C.2.2.2 Poisson Distribution</td><td>295</td></tr><tr><td>C.2.2.3 Gaussian Distribution</td><td>295</td></tr><tr><td>C.2.2.4 Student's t Distribution</td><td>296</td></tr><tr><td>C.2.2.5 Split-Normal Distribution</td><td>296</td></tr><tr><td>C.2.2.6 Lorentzian Distribution</td><td>296</td></tr><tr><td>C.2.3 Drawing random variables from an arbitrary distribution</td><td>296</td></tr><tr><td>C.2.3.1 The Rejection Method</td><td>296</td></tr><tr><td>C.2.3.2 Inverting the CDF</td><td>297</td></tr><tr><td>C.3 Trigonometry</td><td>297</td></tr><tr><td>C.3.1 Transformations</td><td>298</td></tr><tr><td>C.3.1.1 Rotations</td><td>298</td></tr><tr><td>C.3.1.2 Relations Between Functions</td><td>298</td></tr><tr><td>C.3.2 Addition</td><td>298</td></tr><tr><td>C.3.2.1 Summing Angles</td><td>298</td></tr><tr><td>C.3.2.2 Inverse Relations</td><td>298</td></tr><tr><td>C.3.3 Linearization</td><td>298</td></tr><tr><td>C.3.3.1 Squares and Cubes</td><td>298</td></tr><tr><td>C.3.3.2 Inverse Relations</td><td>299</td></tr><tr><td><strong>Acknowledgements</strong></td><td><strong>301</strong></td></tr><tr><td><strong>Bibliography</strong></td><td><strong>303</strong></td></tr></tbody></table>

<figure><img src="image_13.png" /></figure>
<figure><img src="image_14.png" /></figure>

# List of Figures

<table><tbody><tr><td>I</td><td>Propaedeutics in Dust Physics</td><td>1</td></tr><tr><td>I.1</td><td>Shape of s, p, d, f electron orbitals</td><td>5</td></tr><tr><td>I.2</td><td>Periodic table of elements</td><td>5</td></tr><tr><td>I.3</td><td>First ionization potentials and electron affinities</td><td>6</td></tr><tr><td>I.4</td><td>Hybrid orbitals of carbon</td><td>8</td></tr><tr><td>I.5</td><td>Different types of molecular bonds</td><td>8</td></tr><tr><td>I.6</td><td>An example of σ and π bonds</td><td>9</td></tr><tr><td>I.7</td><td>Origin of the band structure of a solid</td><td>11</td></tr><tr><td>I.8</td><td>The Fermi level and the different types of solid</td><td>12</td></tr><tr><td>I.9</td><td>Structure of interstellar dust candidates</td><td>14</td></tr><tr><td>I.10</td><td>Appearance of various minerals</td><td>14</td></tr><tr><td>I.11</td><td>Absorption, scattering and emission</td><td>16</td></tr><tr><td>I.12</td><td>Effect of an electromagnetic wave on a dielectric</td><td>16</td></tr><tr><td>I.13</td><td>Amplitude of a forced harmonic oscillator</td><td>18</td></tr><tr><td>I.14</td><td>Idealized optical constants</td><td>20</td></tr><tr><td>I.15</td><td>Molecular transitions</td><td>23</td></tr><tr><td>I.16</td><td>Dielectric functions of different materials</td><td>25</td></tr><tr><td>I.17</td><td>Illustration of the extinction paradox</td><td>27</td></tr><tr><td>I.18</td><td>Mie, Rayleigh and geometric optics regime</td><td>28</td></tr><tr><td>I.19</td><td>Cross-sections of silicate and graphite grains</td><td>29</td></tr><tr><td>I.20</td><td>Oblate and prolate spheroids</td><td>31</td></tr><tr><td>I.21</td><td>Absorption cross-sections of grain aggregates computed with DDA</td><td>32</td></tr><tr><td>I.22</td><td>Stokes parameters</td><td>33</td></tr><tr><td>I.23</td><td>The three types of grain-induced polarization of light</td><td>35</td></tr><tr><td>I.24</td><td>Phonon modes</td><td>37</td></tr><tr><td>I.25</td><td>Heat capacities</td><td>38</td></tr><tr><td>I.26</td><td>Emissivity of grains at thermal equilibrium with the radiation field</td><td>39</td></tr><tr><td>I.27</td><td>Planck averages of graphite and silicates</td><td>41</td></tr><tr><td>I.28</td><td>Diffuse Galactic ISRF</td><td>41</td></tr><tr><td>I.29</td><td>Grain equilibrium temperatures</td><td>42</td></tr><tr><td>I.30</td><td>Temperature fluctuations of grains with different radii</td><td>44</td></tr><tr><td>I.31</td><td>Temperature fluctuations of grains with different starlight intensities</td><td>45</td></tr><tr><td>I.32</td><td>Stochastically heated grains</td><td>47</td></tr><tr><td>I.33</td><td>Transition radius for stochastically heated grains</td><td>48</td></tr><tr><td>I.34</td><td>Photon and electron energy densities</td><td>50</td></tr><tr><td>I.35</td><td>Collisional heating rate</td><td>50</td></tr></tbody></table>

<table><tbody><tr><td>II</td><td>Dust Observables and Models</td><td>51</td></tr><tr><td>II.1</td><td>Early ISM and IR astronomy</td><td>54</td></tr><tr><td>II.2</td><td>Absorbance of Earth's atmosphere</td><td>56</td></tr></tbody></table>

<figure><img src="image_15.png" /></figure>
<figure><img src="image_16.png" /></figure>

<table><tr><td>II.3</td><td>Airborne observatories</td><td>57</td></tr><tr><td>II.4</td><td>All sky maps</td><td>60</td></tr><tr><td>II.5</td><td>Analysis of the Stardust mission</td><td>61</td></tr><tr><td>II.6</td><td>First evidences of interstellar dust</td><td>63</td></tr><tr><td>II.7</td><td>The pioneers</td><td>64</td></tr><tr><td>II.8</td><td>First detection of UIBs</td><td>65</td></tr><tr><td>II.9</td><td>Désert et al. (1990) dust model</td><td>67</td></tr><tr><td>II.10</td><td>Panchromatic dust observables</td><td>70</td></tr><tr><td>II.11</td><td>Galactic extinction curves</td><td>72</td></tr><tr><td>II.12</td><td>MIR extinction</td><td>73</td></tr><tr><td>II.13</td><td>X-ray edges</td><td>74</td></tr><tr><td>II.14</td><td>Polarized extinction and DIBs</td><td>75</td></tr><tr><td>II.15</td><td>Galactic diffuse ISM SED</td><td>76</td></tr><tr><td>II.16</td><td>Solar abundances</td><td>79</td></tr><tr><td>II.17</td><td>Depletion variations within the MW</td><td>80</td></tr><tr><td>II.18</td><td>MW dust composition inferred from depletions</td><td>81</td></tr><tr><td>II.19</td><td>Presolar grains in meteorites</td><td>82</td></tr><tr><td>II.20</td><td>Micrometeorite collection in Antarctica</td><td>83</td></tr><tr><td>II.21</td><td>NASA Ames PAH experiment</td><td>84</td></tr><tr><td>II.22</td><td>Laboratory measurement of silicate opacities</td><td>84</td></tr><tr><td>II.23</td><td>Size distribution of several dust models</td><td>87</td></tr><tr><td>II.24</td><td>Model opacity and emissivity</td><td>90</td></tr><tr><td>II.25</td><td>THEMIS fit of the Galactic constraints</td><td>91</td></tr><tr><td>II.26</td><td>IR approximation of the opacity</td><td>93</td></tr><tr><td>II.27</td><td>Effect of U on the SED</td><td>95</td></tr></table>

## III The Grain Properties of Nearby Galaxies 97

<table><tr><td>III.1</td><td>Moments of the specific intensity</td><td>102</td></tr><tr><td>III.2</td><td>The radiative transfer equation</td><td>104</td></tr><tr><td>III.3</td><td>Solution to the radiative transfer equation for an isothermal cloud, without scattering</td><td>105</td></tr><tr><td>III.4</td><td>Escaping radiation from a spherical cloud</td><td>107</td></tr><tr><td>III.5</td><td>Escaping SED from clumpy spherical clouds</td><td>110</td></tr><tr><td>III.6</td><td>Principle of Monte-Carlo radiative transfer</td><td>112</td></tr><tr><td>III.7</td><td>Drawing photons in a Monte-Carlo radiative transfer model</td><td>113</td></tr><tr><td>III.8</td><td>Drawing scattering angles in a Monte-Carlo radiative transfer model</td><td>113</td></tr><tr><td>III.9</td><td>Spatial distributions of the clumpy radiative transfer model</td><td>114</td></tr><tr><td>III.10</td><td>Random photon path within a clumpy medium</td><td>115</td></tr><tr><td>III.11</td><td>Total SED from the clumpy radiative transfer model</td><td>115</td></tr><tr><td>III.12</td><td>MBB fitting</td><td>117</td></tr><tr><td>III.13</td><td>Phenomenological mixing of physical conditions</td><td>119</td></tr><tr><td>III.14</td><td>Effect of the ISRF hardness on the SED</td><td>120</td></tr><tr><td>III.15</td><td>Degeneracy of the grain size and ISRF distributions</td><td>121</td></tr><tr><td>III.16</td><td>SED fit with the Draine & Li (2007) starlight intensity distribution</td><td>122</td></tr><tr><td>III.17</td><td>Comparison of different SED models</td><td>123</td></tr><tr><td>III.18</td><td>Stellar isochrones</td><td>125</td></tr><tr><td>III.19</td><td>Evolution of stellar SEDs as a function of time</td><td>126</td></tr><tr><td>III.20</td><td>Multiphase SEDs of galaxies</td><td>127</td></tr><tr><td>III.21</td><td>Matryoshka effect demonstrated on the LMC</td><td>129</td></tr></table>

<figure><img src="image_17.png" /></figure>
<table><tbody><tr><td>III.22 Hubble-de-Vaucouleurs galaxy morphology diagram</td><td>130</td></tr><tr><td>III.23 Visible-range image of three nearby galaxies</td><td>132</td></tr><tr><td>III.24 Principal sources of contaminations encountered when modeling SEDs</td><td>134</td></tr><tr><td>III.25 Radiative transfer modeling of NGC 4565</td><td>136</td></tr><tr><td>III.26 Dust mass discrepancy in the LMC</td><td>137</td></tr><tr><td>III.27 Effect of a blast wave on the grain size distribution</td><td>139</td></tr><tr><td>III.28 Grain size distribution in four dwarf galaxies</td><td>140</td></tr><tr><td>III.29 Extinction curves of low-metallicity environments</td><td>141</td></tr><tr><td>III.30 MIR spectra of galaxies</td><td>143</td></tr><tr><td>III.31 Vibrational modes of PAHs</td><td>144</td></tr><tr><td>III.32 Laboratory and theoretical PAH spectra</td><td>145</td></tr><tr><td>III.33 Empirical calibration of UIB profiles</td><td>148</td></tr><tr><td>III.34 MIR spectral fitting methods</td><td>149</td></tr><tr><td>III.35 PAH and small grain templates</td><td>151</td></tr><tr><td>III.36 Diversity of MIR spectra among and within galaxies</td><td>152</td></tr><tr><td>III.37 PAH band ratio correlations inside and among galaxies</td><td>153</td></tr><tr><td>III.38 Theoretical MIR band ratio variations</td><td>154</td></tr><tr><td>III.39 Calibration of PAH ratio diagnostics</td><td>156</td></tr><tr><td>III.40 PAH band ratio as diagnostics of the physical conditions</td><td>157</td></tr><tr><td>III.41 Effect of ISRF hardness on PAH strength</td><td>158</td></tr><tr><td>III.42 Effect of metallicity on the PAH strength</td><td>159</td></tr><tr><td>III.43 Submillimeter excess in NGC 1569</td><td>160</td></tr><tr><td>III.44 Spatially-resolved submm excess in the LMC</td><td>162</td></tr><tr><td>III.45 AKARI 9 µm map of λ-Orionis</td><td>164</td></tr><tr><td>III.46 AME correlation with ionized PAHs in λ-Orionis</td><td>165</td></tr><tr><td>III.47 Cooling function of the ISM at Solar metallicity</td><td>167</td></tr><tr><td>III.48 Photoelectric heating in PDRs</td><td>168</td></tr><tr><td>III.49 Neutral ISM phase diagram</td><td>169</td></tr><tr><td>III.50 H<sub>2</sub> formation on grain surface</td><td>170</td></tr><tr><td>III.51 Structure of a PDR</td><td>172</td></tr><tr><td>III.52 Comparison of visual extinctions in 30 Doradus</td><td>174</td></tr><tr><td>III.53 Metallicity effect on the CO-dark gas</td><td>175</td></tr><tr><td>III.54 Molecular gas pressure in the center of M 83</td><td>176</td></tr></tbody></table>

# IV Modeling Cosmic Dust Evolution 177

<table><tbody><tr><td>IV.1 The interstellar dust lifecycle</td><td>179</td></tr><tr><td>IV.2 Average nuclear binding energies per nucleon</td><td>181</td></tr><tr><td>IV.3 Schematic representation of stellar evolution</td><td>184</td></tr><tr><td>IV.4 Initial mass functions</td><td>185</td></tr><tr><td>IV.5 Parametric star formation histories</td><td>186</td></tr><tr><td>IV.6 Nucleosynthesis origin of the main elements</td><td>187</td></tr><tr><td>IV.7 Solar metallicity elemental stellar yields</td><td>187</td></tr><tr><td>IV.8 Spatially-resolved SED fits in N 44 and N 66</td><td>191</td></tr><tr><td>IV.9 Dust-to-gas mass surface density relation in N 44 and N 66</td><td>192</td></tr><tr><td>IV.10 Grain growth timescales</td><td>194</td></tr><tr><td>IV.11 Lifetimes of small grains in a radiation field</td><td>197</td></tr><tr><td>IV.12 Carving out of PAHs by UV photons in N 11</td><td>197</td></tr><tr><td>IV.13 Thermal and kinetic sputtering times of silicates and carbon grains</td><td>198</td></tr><tr><td>IV.14 Evidence of thermal sputtering in elliptical galaxies</td><td>199</td></tr></tbody></table>
<table><tr><td>IV.15</td><td>Effects of dust evolution on the SEDs of galaxies.</td><td>201</td></tr><tr><td>IV.16</td><td>Dust evolution tracks for a MW-like galaxy</td><td>205</td></tr><tr><td>IV.17</td><td>Effects of SFH-related parameters on dust evolution</td><td>207</td></tr><tr><td>IV.18</td><td>Effects of tuning parameters on dust evolution</td><td>208</td></tr><tr><td>IV.19</td><td>Dustiness-metallicity relation fitted with a dust evolution model</td><td>209</td></tr><tr><td>IV.20</td><td>Empirical estimates of dust evolution timescales as a function of metallicity</td><td>211</td></tr><tr><td>IV.21</td><td>Evolution of the mass fraction of small a-C(:H) grains with metallicity and starlight intensity.</td><td>214</td></tr><tr><td>IV.22</td><td>The potential of quiescent very-low-metallicity galaxies to understand the origin of small a-C(:H) grains</td><td>215</td></tr></table>

V Methodological Effort and Epistemological Reflection 217

<table><tr><td>V.1</td><td>Venn diagram to demonstrate Bayes' rule</td><td>221</td></tr><tr><td>V.2</td><td>Simulation of the measure of a stellar flux to compare Bayesian and frequentist methods.</td><td>223</td></tr><tr><td>V.3</td><td>Bayesian and frequentist solutions to the problem of Fig. V.2</td><td>225</td></tr><tr><td>V.4</td><td>The benefits of using an informative prior</td><td>226</td></tr><tr><td>V.5</td><td>Flux measures with a non-linear detector</td><td>228</td></tr><tr><td>V.6</td><td>Markov Chain Monte-Carlo algorithms</td><td>231</td></tr><tr><td>V.7</td><td>Importance of the choice of the Metropolis-Hastings proposal distribution</td><td>234</td></tr><tr><td>V.8</td><td>Post-processing of the MCMC of a sample of sources</td><td>235</td></tr><tr><td>V.9</td><td>MCMC statistics of a sample of sources</td><td>236</td></tr><tr><td>V.10</td><td>Demonstration of the use of posterior predictive p-values</td><td>237</td></tr><tr><td>V.11</td><td>Bayesian and frequentist hypothesis testing</td><td>239</td></tr><tr><td>V.12</td><td>The probability pioneers</td><td>244</td></tr><tr><td>V.13</td><td>The frequentist promoters</td><td>246</td></tr><tr><td>V.14</td><td>The Bayesian resistance</td><td>247</td></tr><tr><td>V.15</td><td>Three figures of modern epistemology</td><td>248</td></tr><tr><td>V.16</td><td>Bayes factors and parsimony</td><td>252</td></tr><tr><td>V.17</td><td>Example of hierarchical Bayesian SED fits</td><td>259</td></tr><tr><td>V.18</td><td>Posterior distributions with standard and hierarchical Bayesian methods</td><td>260</td></tr><tr><td>V.19</td><td>Comparison of least-squares, standard Bayesian and HB methods</td><td>261</td></tr><tr><td>V.20</td><td>Solving the emissivity-index-temperature degeneracy of MBBs with a HB model</td><td>262</td></tr><tr><td>V.21</td><td>Demonstration of the effect of the prior in a HB model</td><td>263</td></tr><tr><td>V.22</td><td>The holistic approach: inclusion of external parameters into the prior</td><td>264</td></tr></table>

VI Conclusion and Prospective 267

A List of Acronyms 275

<table><tr><td>A.1</td><td>Spectral domains represented over the SED of a nearby galaxy</td><td>279</td></tr></table>

B Astronomers and Units 281

C Useful Formulae 289
<table><tr><td>C.1</td><td>Most common coordinate systems</td><td>290</td></tr><tr><td>C.2</td><td>Two ways of slicing the πs</td><td>292</td></tr><tr><td>C.3</td><td>Methods for drawing random numbers from arbitrary distributions</td><td>297</td></tr><tr><td>D.1</td><td>Nerdy allegory of my collaboration network</td><td>301</td></tr></table>
# List of Tables

<table><tbody><tr><td>I</td><td>Propaedeutics in Dust Physics</td><td>1</td></tr><tr><td>I.1</td><td>Atomic quantum numbers</td><td>3</td></tr><tr><td>I.2</td><td>Atomic shell structure</td><td>4</td></tr><tr><td>I.3</td><td>General properties of interstellar dust candidates</td><td>15</td></tr><tr><td>II</td><td>Dust Observables and Models</td><td>51</td></tr><tr><td>II.1</td><td>Chronology of the main ISD breakthroughs</td><td>69</td></tr><tr><td>II.2</td><td>Comparison between different dust models</td><td>86</td></tr><tr><td>II.3</td><td>Moments of the THEMIS size distribution</td><td>92</td></tr><tr><td>II.4</td><td>Dustiness and other ratios for the THEMIS model</td><td>92</td></tr><tr><td>II.5</td><td>Optical properties of the THEMIS model</td><td>94</td></tr><tr><td>II.6</td><td>Transition wavelengths between small and large grain emission</td><td>94</td></tr><tr><td>II.7</td><td>Emissivity of the THEMIS model</td><td>95</td></tr><tr><td>III</td><td>The Grain Properties of Nearby Galaxies</td><td>97</td></tr><tr><td>III.1</td><td>Mean free path as a function of wavelength and density</td><td>103</td></tr><tr><td>III.2</td><td>Basic properties of the main stellar classes</td><td>124</td></tr><tr><td>III.3</td><td>Free emissivity index MBB fits of nearby galaxies by the Planck collaboration</td><td>136</td></tr><tr><td>III.4</td><td>UIB profile parameters</td><td>147</td></tr><tr><td>III.5</td><td>Most prominent MIR gas lines</td><td>150</td></tr><tr><td>III.6</td><td>Phases of the ISM</td><td>166</td></tr><tr><td>IV</td><td>Modeling Cosmic Dust Evolution</td><td>177</td></tr><tr><td>IV.1</td><td>IMF properties</td><td>184</td></tr><tr><td>IV.2</td><td>Summary of the observational evidence about interstellar dust origin at normal metallicity</td><td>212</td></tr><tr><td>V</td><td>Methodological Effort and Epistemological Reflection</td><td>217</td></tr><tr><td>V.1</td><td>Jeffreys strength of evidence scale</td><td>240</td></tr><tr><td>V.2</td><td>Pros and cons of the Bayesian and frequentist methods</td><td>242</td></tr><tr><td>V.3</td><td>Comparison of Bayesian and Popperian epistemologies</td><td>253</td></tr><tr><td>V.4</td><td>Inferred statistical properties of the HB model in Fig. V.19.c</td><td>261</td></tr><tr><td>VI</td><td>Conclusion and Prospective</td><td>267</td></tr><tr><td>A</td><td>List of Acronyms</td><td>275</td></tr></tbody></table>
<figure><img src="image_18.png" /></figure>
<figure><img src="image_19.png" /></figure>

Abstract

*The truth is the whole.*

(George Wilhelm Friedrich HEGEL; Hegel, 1807)

Interstellar dust is a key physical ingredient of galaxies, obscuring star formation, regulating the heating and cooling of the gas, and building-up chemical complexity. In this manuscript, I give a wide review of interstellar dust properties and some of the modern techniques used to study it. I start with a general introduction presenting the main concepts, in molecular and solid-state physics, required to understand the contemporary literature on the subject. I then review the empirical evidence we currently use to constrain state-of-the-art dust models. Follows a long discussion about our current understanding of the grain properties of nearby galaxies, with an emphasis on the results from spectral energy distribution modeling. The following chapter presents dust evolution at all scales. I review the different microphysical evolution processes, and the way they are accounted for in cosmic dust evolution models. I give my take on the origin of interstellar dust in galaxies of different metallicities. The last chapter focusses on methodology. I give an introduction to the Bayesian method and compare it to frequentist techniques. I discuss the epistemological consequences of the two approaches, and show why the field of interstellar dust requires a probabilistic viewpoint. I end the manuscript with a summary of the major breakthroughs achieved in the past decade, and delineate a few perspectives for the next decade.

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /></figure>
<figure><img src="image_22.png" /></figure>

# Introduction

*The unity of all science consists alone in its method, not in its material.*

(Karl PEARSON; Pearson, 1892)

## Interstellar Dust: A Key To Understanding Galaxy Evolution

Understanding galaxy evolution is one of the main objectives of observational cosmology, as it allows mapping the history of the Universe, from the dark ages to the present times (e.g. Madau & Dickinson, 2014; Buat, 2015). At the center of this evolution lies the *Interstellar Medium* (ISM). This complex intertwining of ionized, atomic and molecular gas phases mixed with dust grains, fills the volume of a galaxy, ultimately leading to star formation (SF), by gravitational collapse (e.g. Klessen & Glover, 2016). Although accounting for only ≈ 1% of its mass, dust is an essential component of the ISM. It consists of solid particles (0.3 nm ≲ radius ≲ 0.3 µm) made out of the available heavy elements, predominantly arranged in silicate and carbonaceous compounds (e.g. Draine, 2003a). These grains, absorbing and scattering starlight, have a radical impact on a galaxy.

* They have a nefarious role of obscuring our direct view of star formation. They normally re-radiate ≈ 25% of the stellar power in the *InfraRed* (IR; cf. Table A.4), and up to ≈ 99% in ultraluminous IR galaxies (e.g. Bianchi et al., 2018; Clements et al., 1996).

* Dust is an essential ingredient of star formation, as it contributes to (e.g. Li & Greenberg, 2003):

* radiatively evacuating the gravitational energy of collapsing molecular clouds;

* shielding the molecules from starlight, which protect them from destruction and reduces their ionization fraction, allowing the formation of protostellar cores.

* In addition, grains are responsible for the heating of the gas in *PhotoDissociation Regions* (PDR), by the photoelectric effect (Draine, 1978; Kimura, 2016).

* They are also catalysts of numerous chemical reactions, including the formation of the most abundant molecule in the Universe, H₂ (Gould & Salpeter, 1963; Bron et al., 2014).

* Elongated grains tend to align with the magnetic field. Polarized extinction (in the visible) and emission (in the IR) by grains are therefore one of the most popular tools to study the magnetic field in the ISM (e.g. Planck Collaboration et al., 2016b,d; Guillet et al., 2018).

☞ A detailed knowledge of the dust properties and their evolution is therefore imperative in order to both interpret observations of galaxies and model their ISM.

Dust physics is characterized by the great complexity of its make-up, as the number of ways to combine elements to build interstellar solids is virtually limitless. Most of the progress in this field thus relies on empirical constraints: observations and laboratory experiments on cosmic dust analogs. Our current knowledge of *interstellar dust* (ISD) properties is however hampered by several factors (Galliano et al., 2018, for a review). First, observations of interstellar regions are always the superimposition, along the line of sight and within the telescope beam, of a range of physical conditions: (i) intensity and hardness of the *Interstellar Radiation Field* (ISRF); (ii) gas density; and

<figure><img src="image_23.png" /></figure>
<figure><img src="image_24.png" /></figure>

(iii) presence of shocks. Consequently, since we can never accurately recover the 3-Dimensional (3D) structure of a region, several degeneracies between the grain constitution and their excitation prevent a unique solution. Second, the grain constitution<fnref n="1" /> is known to evolve under the effects of ISRF and gas density (e.g. Draine, 2009; Jones et al., 2013; Ysard et al., 2015). It is thus likely that, in addition to variations of excitation conditions, ISD observables are coming from a combination of altered grain mixtures. Finally, the derivation of precise dust properties, even from observations towards a uniform, uncontaminated region, is limited by an incomplete spectral coverage and by instrumental uncertainties.

It follows that a rigorous attempt at quantifying grain parameters and their evolution must account for these factors in both the choices of astrophysical targets and modeling approach.

# The Relevance of Nearby Galaxies

Due to their proximity, ISM regions of our own galaxy, the Milky Way (MW), can be observed with the finest linear resolution. MW studies have consequently laid the ground for the development of physical dust models (Draine, 2003a, for a review). They are however limited by the small range of environmental conditions they span.

* There are no really massive star-forming regions in the MW. The only Super Star Cluster (SSC; M★ ≳ 10⁵ M☉; e.g. Johnson, 2001; Dowell et al., 2008) is Westerlund 2 in RCW 49 (Moffat et al., 1991).

* As in most spiral galaxies, there is a radial gradient of metallicity (the mass fraction of elements heavier than He, noted Z; e.g. Asplund et al., 2009). It however ranges narrowly (0.7 Z☉ ≲ Z ≲ 2 Z☉; Henry & Worthey, 1999).

* The massive central black hole, Sgr A* of the Galactic center is relatively passive (Mezger et al., 1996) compared to Active Galactic Nuclei (AGN), such as NGC 1068 (e.g. Le Floc'h et al., 2001).

MW studies are also limited by the confusion along the sightline, as we are seeing the projected material of the entire disk. Finally, distances of interstellar clouds can be difficult to estimate for the same reason, although 3D maps of the MW are becoming more precise (e.g. Lallemand et al., 2018).

In contrast, nearby galaxies (closer than ≈ 100 Mpc; Galliano et al., 2018) represent an undertapped population with several potentials. First, they harbor a wider range of environmental parameters, allowing us, in particular, to probe dust in extreme conditions.

* Nearby galaxies, especially Blue Compact Dwarfs (BCD), contain numerous SSCs (e.g. O'Connell et al., 1994; Johnson et al., 2000; Martín-Hernández et al., 2005). They are ideal laboratories to understand the impact of massive star formation on the ISM.

* Extremely low-metallicity objects, such as I Zw 18 (Z ≈ 1/35 Z☉; Izotov et al., 1999), allow us to study dust in environments where the chemical enrichment resembles primordial galaxies.

* AGNs have a radical impact on the ISM of their host galaxies, heavily processing the grains (e.g. their crystalline fraction; Spoon et al., 2006). In addition, bright AGNs can be used to study grains in absorption (e.g. Spoon et al., 2002; Mason et al., 2007).

Second, face-on galaxies, observed at high Galactic latitude, provide clearer sightlines than in the MW. Finally, the lower linear resolution we can reach in nearby objects (≈ 100 pc to 1 kpc in the IR) is the ideal length scale to adopt a statistical description of the distribution of clouds and stars<fnref n="2" />, whereas detailed, parsec-scale MW studies are left to the nearly impossible task of inferring the precise geometry of each single cloud and the position in space of the surrounding stars.

<figure><img src="image_25.png" /></figure>

<footnote n="1">1. Stoichiometry, chemical composition, solid-state structure, size distribution and abundance relative to the gas.</footnote>

<footnote n="2">2. The average distance between stars is ≈ 1 pc, and the typical size of molecular clouds ranges from ≈ 1 to ≈ 100 pc (e.g. Solomon et al., 1987).</footnote>
<figure><img src="image_26.png" /></figure>

# Scope of the Manuscript

Most of my career until now has been devoted to studying the dust properties and their evolution in nearby galaxies. I have chosen to focus the present manuscript on the following directions.

* It is primarily a synthesis of my achievements in this field. The goal is not to repeat the content of my publications, but to put them in perspective with other studies. For that reason, the presentation of my results occupy only a small fraction of the manuscript.

* It also provides a review of our current understanding of ISD, and outlines the most promising directions this field should explore during the next decade.

* Finally, I have tried to compile introductory material, concepts and figures that could be useful to students starting in this field, but that are otherwise scattered across the literature. Unless otherwise noted in the caption, the figures presented here are original and are licensed under CC BY-SA 4.0³.

It is divided as follows.

**Chap. I** provides a reminder of the main concepts at the foundation of ISD physics.

**Chap. II** gives a general introduction about the most reliable observational evidences we have about ISD, and the current models attempting at synthesizing them.

**Chap. III** reviews dust properties in the MW and nearby galaxies, and the way they are constrained.

**Chap. IV** reviews evidences of dust evolution and models accounting for their formation and destruction at the scale of a galaxy.

**Chap. V** is a more original take on my methodological approach, motivated by some epistemological concepts.

**Chap. VI** is a summary of what we have learned about ISD in the past decade and what we should do during the next one.

<figure><img src="image_27.png" /></figure>

<footnote n="3">3. This means you can freely reproduce or modify a figure without my permission, as long as you credit my name, by citing this HDR, and give the link to the license.</footnote>
<figure><img src="image_28.png" /></figure>
<figure><img src="image_29.png" /></figure>

# Chapter I

# Propaedeutics in Dust Physics

> As soon as we thought something, look in what sense the opposite is true.

(Simone WEIL; Weil, 1947)

## Contents

<table><tr><td>I.1</td><td>The Make-Up of Solids</td><td>3</td></tr><tr><td>I.1.1</td><td>Atomic Structure</td><td>3</td></tr><tr><td>I.1.1.1</td><td>The Hydrogen Atom</td><td>3</td></tr><tr><td>I.1.1.2</td><td>Polyelectronic Atoms</td><td>3</td></tr><tr><td>I.1.1.3</td><td>The Valence Shell</td><td>4</td></tr><tr><td></td><td>Ionization potential.</td><td>4</td></tr><tr><td></td><td>Electron affinity.</td><td>4</td></tr><tr><td></td><td>Electronegativity.</td><td>4</td></tr><tr><td>I.1.1.4</td><td>Orbital Hybridisation</td><td>7</td></tr><tr><td>I.1.2</td><td>Molecular Bonding</td><td>7</td></tr><tr><td>I.1.2.1</td><td>Covalent Bonds</td><td>7</td></tr><tr><td>I.1.2.2</td><td>Ionic Bonds</td><td>9</td></tr><tr><td>I.1.2.3</td><td>Metallic Bonds</td><td>10</td></tr><tr><td>I.1.2.4</td><td>Intermolecular Attraction</td><td>10</td></tr><tr><td>I.1.3</td><td>The Solid State</td><td>10</td></tr><tr><td>I.1.3.1</td><td>The Different Types of Solids</td><td>10</td></tr><tr><td>I.1.3.2</td><td>The Band Structure of Solids</td><td>10</td></tr><tr><td>I.1.3.3</td><td>The Fermi Level</td><td>11</td></tr><tr><td>I.1.4</td><td>Interstellar Dust Candidates</td><td>12</td></tr><tr><td>I.1.4.1</td><td>Silicates</td><td>12</td></tr><tr><td></td><td>Olivine.</td><td>13</td></tr><tr><td></td><td>Pyroxene.</td><td>13</td></tr><tr><td>I.1.4.2</td><td>Hydrogenated Amorphous Carbon</td><td>13</td></tr><tr><td>I.1.4.3</td><td>Graphite</td><td>15</td></tr><tr><td>I.1.4.4</td><td>Polycyclic Aromatic Hydrocarbons (PAH)</td><td>15</td></tr><tr><td>I.2</td><td>The Interaction of Light with Solids</td><td>15</td></tr><tr><td>I.2.1</td><td>Bonds as Harmonic Oscillators</td><td>17</td></tr><tr><td>I.2.1.1</td><td>The Harmonic Oscillator Amplitude</td><td>17</td></tr><tr><td>I.2.1.2</td><td>The Plasma Frequency</td><td>17</td></tr><tr><td>I.2.1.3</td><td>The Dielectric Function</td><td>18</td></tr><tr><td>I.2.1.4</td><td>Harmonic Oscillator Cross-Section</td><td>19</td></tr></table>

<figure><img src="image_30.png" /></figure>
<table>
  <tbody>
    <tr>
      <td>I.2.1.5</td>
      <td>Optical Constants of Conductors</td>
      <td>21</td>
    </tr>
    <tr>
      <td>I.2.1.6</td>
      <td>The Kramers-Kronig Relations</td>
      <td>22</td>
    </tr>
    <tr>
      <td></td>
      <td>Implications.</td>
      <td>22</td>
    </tr>
    <tr>
      <td></td>
      <td>Interpretation.</td>
      <td>22</td>
    </tr>
    <tr>
      <td></td>
      <td>Constraint on the Cross-Section.</td>
      <td>22</td>
    </tr>
    <tr>
      <td>I.2.2</td>
      <td>Grain Optical Properties</td>
      <td>23</td>
    </tr>
    <tr>
      <td>I.2.2.1</td>
      <td>Why Are Most Dust Features in the MIR?</td>
      <td>23</td>
    </tr>
    <tr>
      <td>I.2.2.2</td>
      <td>Dielectric Functions of Realistic Materials</td>
      <td>23</td>
    </tr>
    <tr>
      <td>I.2.2.3</td>
      <td>Computing Grain Cross-Sections</td>
      <td>24</td>
    </tr>
    <tr>
      <td></td>
      <td>Geometrical optics.</td>
      <td>26</td>
    </tr>
    <tr>
      <td></td>
      <td>The Mie regime.</td>
      <td>26</td>
    </tr>
    <tr>
      <td></td>
      <td>The Rayleigh regime.</td>
      <td>27</td>
    </tr>
    <tr>
      <td>I.2.2.4</td>
      <td>Beyond Homogeneous Spheres</td>
      <td>30</td>
    </tr>
    <tr>
      <td></td>
      <td>Effective medium theory (EMT).</td>
      <td>30</td>
    </tr>
    <tr>
      <td></td>
      <td>Ellipsoids in the Rayleigh regime.</td>
      <td>31</td>
    </tr>
    <tr>
      <td></td>
      <td>Discrete Dipole Approximation (DDA).</td>
      <td>31</td>
    </tr>
    <tr>
      <td>I.2.2.5</td>
      <td>Polarization</td>
      <td>32</td>
    </tr>
    <tr>
      <td></td>
      <td>The Stokes parameters.</td>
      <td>32</td>
    </tr>
    <tr>
      <td></td>
      <td>Grain alignment.</td>
      <td>33</td>
    </tr>
    <tr>
      <td></td>
      <td>Polarization by scattering.</td>
      <td>33</td>
    </tr>
    <tr>
      <td></td>
      <td>Dichroic extinction.</td>
      <td>34</td>
    </tr>
    <tr>
      <td></td>
      <td>Polarized emission.</td>
      <td>34</td>
    </tr>
    <tr>
      <td>I.2.3</td>
      <td>Heat Capacities</td>
      <td>34</td>
    </tr>
    <tr>
      <td>I.2.3.1</td>
      <td>Distribution of Harmonic Oscillators</td>
      <td>34</td>
    </tr>
    <tr>
      <td>I.2.3.2</td>
      <td>Debye's Model</td>
      <td>36</td>
    </tr>
    <tr>
      <td>I.2.3.3</td>
      <td>Heat Capacities of Realistic Materials</td>
      <td>37</td>
    </tr>
    <tr>
      <td>I.2.4</td>
      <td>Heating and Cooling</td>
      <td>37</td>
    </tr>
    <tr>
      <td>I.2.4.1</td>
      <td>Kirchhoff's Law</td>
      <td>37</td>
    </tr>
    <tr>
      <td></td>
      <td>Opacity.</td>
      <td>38</td>
    </tr>
    <tr>
      <td></td>
      <td>Emissivity.</td>
      <td>38</td>
    </tr>
    <tr>
      <td></td>
      <td>Modified Black Body (MBB).</td>
      <td>39</td>
    </tr>
    <tr>
      <td>I.2.4.2</td>
      <td>Equilibrium Heating</td>
      <td>40</td>
    </tr>
    <tr>
      <td></td>
      <td>Planck average.</td>
      <td>40</td>
    </tr>
    <tr>
      <td></td>
      <td>ISRF-averaged efficiency.</td>
      <td>40</td>
    </tr>
    <tr>
      <td></td>
      <td>The diffuse ISRF.</td>
      <td>40</td>
    </tr>
    <tr>
      <td></td>
      <td>Equilibrium temperatures.</td>
      <td>42</td>
    </tr>
    <tr>
      <td>I.2.4.3</td>
      <td>Stochastic Heating</td>
      <td>42</td>
    </tr>
    <tr>
      <td></td>
      <td>Absorption and cooling times.</td>
      <td>42</td>
    </tr>
    <tr>
      <td></td>
      <td>The temperature fluctuations.</td>
      <td>43</td>
    </tr>
    <tr>
      <td></td>
      <td>Out-of-equilibrium emission.</td>
      <td>43</td>
    </tr>
    <tr>
      <td></td>
      <td>Numerical methods.</td>
      <td>46</td>
    </tr>
    <tr>
      <td></td>
      <td>Equilibrium criterion.</td>
      <td>46</td>
    </tr>
    <tr>
      <td>I.2.4.4</td>
      <td>Collisionnal Heating</td>
      <td>48</td>
    </tr>
    <tr>
      <td></td>
      <td>Electronic heating rate.</td>
      <td>48</td>
    </tr>
    <tr>
      <td></td>
      <td>Cases where collisions dominate the heating.</td>
      <td>49</td>
    </tr>
  </tbody>
</table>

Most reviews about ISD (e.g. Draine, 2003a; Whittet, 2003; Tielens, 2005; Draine, 2011; Jones, 2016a,b,c; Galliano et al., 2018) assume that the reader has a good knowledge of solid-state physics<fnref n="1" />. It is however not always the case, especially among students. The present chapter is intended to synthesize

<footnote n="1">1. The book by Krügel (2003) is a notable exception. It starts from elementary electrodynamics and atomic physics.</footnote>
<figure><img src="image_31.png" /></figure>

the basic knowledge necessary to understand contemporary publications in the field. We have opted for a simple presentation of the most important concepts, accompanied with a few original figures. We present the most important formulae and refer the reader to reliable textbooks for their proofs. We have tried to answer everything students always wanted to know about dust, but were afraid to ask.

# I.1 The Make-Up of Solids

Atoms can be combined to form molecules or solids. The properties of these compounds depend greatly on the way their constitutive atoms are bonded together, by their electrons. Electrons being fermions (their spin is 1/2), the *Pauli exclusion principle* implies that each of them must occupy a different state in a system, characterized by its wave function (e.g. Tome II, Chapter XIV of Cohen-Tannoudji et al., 1996). The electronic shells of atoms, the bonds of molecules and the band structure of solids all follow from this principle.

## I.1.1 Atomic Structure

### I.1.1.1 The Hydrogen Atom

The electrons of an atom each have a distinct wave function, $\Psi$, called *orbital*. The probability density function of presence of the electron is proportional to $|\Psi|^2$. These orbitals are solutions to the Schrödinger equation, in the electrostatic well created by the nucleus, assumed to be infinitely heavy<fnref n="2" />. These solutions are relatively simple in the case of the hydrogen atom (cf. e.g. Chap. 3 of Bransden & Joachain, 1983). Its single electron can occupy different shells, characterized by their *principal quantum number, n*. This quantum number determines the energy level of the orbitals ($E \propto 1/n^2$) as well as their size (cf. Table I.1). Each individual shell is divided into subshells, characterized by the *azimuthal number, l*, quantifying the angular momentum of the electron ($L \propto \sqrt{l(l+1)}$), which can be 0, for s subshells (cf. Table I.2). The orbitals in each subshell are combinations of spherical harmonic functions. These functions are anisotropic. The *magnetic quantum number, m*<sub>l</sub>, quantifies their orientation in space, as displayed in Fig. I.1. Finally, the *spin quantum number, m*<sub>s</sub>, quantifies the direction of the electronic spin (up or down).

<table><caption>TABLE I.1 – *Atomic quantum numbers*. Adapted from Table 7.2 of Atkins (1992).</caption><thead><tr><td>Name</td><td>Symbol</td><td>Values</td><td>Signification</td></tr></thead><tbody><tr><td>Principal</td><td>n</td><td>1, 2, ..., ∞</td><td>Energy ($E \propto 1/n^2$ for H) or size of the shell</td></tr><tr><td>Azimuthal</td><td>l</td><td>0, 1, ..., n − 1</td><td>Angular momentum ($L \propto \sqrt{l(l+1)}$)</td></tr><tr><td>Magnetic</td><td>m<sub>l</sub></td><td>l, l − 1, ..., −l</td><td>Orientation (spherical harmonic combination)</td></tr><tr><td>Spin</td><td>m<sub>s</sub></td><td>+1/2, −1/2</td><td>Magnetic moment (spin direction)</td></tr></tbody></table>

### I.1.1.2 Polyelectronic Atoms

The different energy levels of an atom can be populated by excitation (collision or photon absorption). In the fundamental state of the H atom, the electron occupies the *n* = 1 level. The electrons of a polyelectronic atom in its fundamental state occupy the lowest energy orbitals available. Each electron must have a unique set of the four quantum numbers (Table I.1). The nucleus charge, Z, is higher, which causes the inner shells to be closer to the nucleus. The electric field seen by the outer shells is thus partially screened by the inner shells. A fundamental difference between polyelectronic

<figure><img src="image_32.png" /></figure>

<footnote n="2">2. The electron-to-proton mass ratio is $m_e/m_p \approx 5 \times 10^{-4}$ (cf. Table B.2).</footnote>
<table><caption>TABLE I.2 – *Atomic shell structure.* We have highlighted the correspondence with the shells of Bohr's pre-quantum atomic model. The letters s, p, d, f are used to label orbitals with different values of *l*.</caption><thead><tr><th rowspan="2">Principal<br>quantum<br>number</th><th rowspan="2">Bohr<br>shells</th><th colspan="4">Azimuthal quantum number</th><th rowspan="2">Subshell letter<br>Number of electrons per subshell</th></tr><tr><th>s</th><th>p</th><th>d</th><th>f</th></tr></thead><tbody><tr><td>n = 1</td><td>K</td><td>l = 0</td><td></td><td></td><td></td><td></td></tr><tr><td>n = 2</td><td>L</td><td>l = 0</td><td>l = 1</td><td></td><td></td><td></td></tr><tr><td>n = 3</td><td>M</td><td>l = 0</td><td>l = 1</td><td>l = 2</td><td></td><td></td></tr><tr><td>n = 4</td><td>N</td><td>l = 0</td><td>l = 1</td><td>l = 2</td><td>l = 3</td><td></td></tr></tbody></table>

atoms and hydrogen is however the mutual repulsion of the electrons (cf. e.g. Chap. 7 of Bransden & Joachain, 1983). The different subshells (s, p, d, f) of a given shell having different geometries, the mutual repulsion depends on *l*. The subshells of a given *n* thus have different energies. The electronic configuration of atoms in their fundamental state results from the ranking in energy of these levels. The possible number of electrons per subshell is given in Table I.2. We have represented the electronic configuration of atoms in the periodic table (Fig. I.2).

### I.1.1.3 The Valence Shell

The outer shell is called the *valence shell*. It contains the electrons responsible for molecular bonds (cf. Sect. I.1.2) and shaping the optical properties of solids (cf. Sect. I.2). We will see that the nature of the chemical bond depends on the tendency of its atoms: (i) to share electrons; (ii) to form *cations*, by losing one or several electrons; or (iii) to form *anions*, by gaining one or several electrons.

**Ionization potential.** Panel (a) of Fig. I.3 shows the *first ionization potential*, $I_1$, of the elements in Fig. I.2. Atoms with a low $I_1$ tend to form stable *cations*. Noble gases have the largest $I_1$ of their row, because their last shell is full. More generally, Fig. I.3 shows that $I_1$ increases from the left to the right of the periodic table (Fig. I.2), as the valence shell gets fuller. This is because, at a given *n*, moving to the right of the table increases the effective nucleus charge $Z-Z_{subshells}$, making the valence shell more tightly bound. It also decreases from the top to the bottom, as the energy level of the outer shell decreases with *n* as $1/n^2$.

Metals tend to form stable *cations*.

**Electron affinity.** Panel (b) of Fig. I.3 shows the *first electron affinity*, $A_1$, of the elements in Fig. I.2. Atoms with a high electron affinity tend to form stable *anions*. $A_1$ follows roughly the same trend as $I_1$, with some exceptions. Noble gases have their last shell full, they therefore tend to remain neutral and have a negative $A_1$. Alkaline earth metals also have negative $A_1$, because of the energy difference between their full *ns* and empty *np* shells.

Non metals tend to form stable *anions*.

**Electronegativity.** The balance between $I_1$ and $A_1$ gives an idea of the tendency of an atom to attract electrons in a bond. Mulliken's *electronegativity*, $\chi$, is defined as (e.g. Huheey et al., 1993):

$$ \chi = 0.187 \times \left( \frac{I_1}{1 \text{ eV}} + \frac{A_1}{1 \text{ eV}} \right) + 0.17. \qquad (I.1) $$

If we except noble gases, we see that $\chi$ will be higher to the right of the periodic table, and will decrease from the top to the bottom.
<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /><figcaption>FIGURE I.1 – *Shape of s, p, d, f electron orbitals.* These shapes represent the surfaces encompassing the region where the electron has a 90% probability of presence. The orbitals are identified by the letter corresponding to the value of *l* (cf. Table I.2), with the value of *m*<sub>l</sub> as an index. Credit: UCDavis Chemwiki, licensed under CC BY-NC-SA 3.0 US.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>FIGURE I.2 – *Periodic table of elements.* We show only the first four rows, as they contain all the elements relevant to ISM physics. Z is the charge and A, the atomic weight. In each cell, we show the electronic configuration in the fundamental state. We do not repeat the part of the configuration corresponding to the noble gas of the previous row ([He], [Ne], [Ar]). We have annotated the table with the four quantum numbers of the last electron. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_36.png" /></figure>
<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

FIGURE I.3 – *First ionization potentials and electron affinities.* In panel (a), we display the first ionization potential. It is the minimum energy required to expel the most loosely bound electron, in the fundamental state. It is the energy lost during the gas phase reaction: $X \rightarrow X^+ + e^-$. In panel (b), we display the first electron affinity, which is the energy gained by adding an electron to the atom. It is the energy released during the gas phase reaction: $X + e^- \rightarrow X^-$. Elements with a negative electron affinity (yellow hatched area) do not form stable anions. We display the different groups with the same color code as in Fig. I.2. Licensed under CC BY-SA 4.0.

<figure><img src="image_39.png" /></figure>
<figure><img src="image_40.png" /></figure>

### I.1.1.4 Orbital Hybridisation

To explain the shape of molecules, the concept of *hybrid orbitals* was introduced in the 1930's by Linus PAULING. The principle is that orbitals with similar energies can be linearly combined together to form new orbitals with different shapes<fnref n="3" />. The most relevant example to ISD is the hybridisation of carbon. A 2s electron can be promoted to 2p, resulting in the following configuration.

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

The carbon is now in an excited state, noted C*. The promotion requires 4.2 eV. From this new state, the following combinations of orbitals are possible.

**sp<fnref n="3" /> hybrids** are 1/4 s and 3/4 p. This hybridisation results in four sp<fnref n="3" /> orbitals arranged in a tetrahedron, shown in Fig. I.4.a. For instance, C in methane (CH₄) is sp<fnref n="3" /> hybridized. The electronic configuration of the *n* = 2 shell becomes the following.

<figure><img src="image_43.png" /></figure>

**sp² hybrids** are 1/3 s and 2/3 p. This hybridisation results in one standard p orbital and three sp² orbitals trigonally-arranged, shown in Fig. I.4.b. For instance, C in benzene (C₆H₆) is sp² hybridized. The electronic configuration of the *n* = 2 shell becomes the following.

<figure><img src="image_44.png" /></figure>

**sp hybrid** is 1/2 s and 1/2 p. This hybridisation results in two standard p orbitals and two sp orbitals linearly-arranged, shown in Fig. I.4.c. For instance, C in acetylene (C₂H₂) is sp hybridized. The electronic configuration of the *n* = 2 shell becomes the following.

<figure><img src="image_45.png" /></figure>

## I.1.2 Molecular Bonding

Chemical bonds are the result of the overlap between the outer orbitals of two atoms whose valence shell is not full (cf. e.g. Chap. 8 of Atkins, 1992). Despite their mutual repulsion, sharing electrons leads to a lower energy state, in which a stable bonded molecule is formed. Covalent, ionic and metallic bonds, that we will define below, typically have dissociation energies of a few eV. The atomic spacing in a molecule or a solid is of the order of a few Å.

### I.1.2.1 Covalent Bonds

A *covalent bond* is formed when a pair of electrons with anti-parallel spin is shared between two atoms of similar electronegativity. The more the orbitals overlap, the stronger the bond is. The electron density is the highest between the two atoms, resulting in a directional bond. For instance, the H₂ (Fig. I.5.a) and CO molecule bonds, as well as the C-C and C-H bonds in hydrocarbons, are all covalent. Covalent bonds are weakly polar. Symmetric molecules such as H₂ are non-polar, whereas asymmetric molecules, such as CO are polar, because of the difference in electronegativity of C and O.

<figure><img src="image_46.png" /></figure>

<footnote n="3">3. The Schrödinger equation is linear. Any combination of solutions is a solution.</footnote>
<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

FIGURE 1.4 – *Hybrid orbitals of carbon.* The left image shows the shape of the four sp³ orbitals of a C atom. They are arranged in a tetrahedral geometry. The center image shows the shape of the three sp² orbitals of a C atom. They are arranged in a trigonal geometry. We have not displayed the remaining p orbital of this C atom. The right image shows the shape of the two sp orbitals of a C atom. They are arranged along a line. We have not displayed the remaining two p orbitals of this C atom. Each bond is represented with a black line indicating its direction: (i) thin solid lines show bonds within the plane of the image; (ii) the thick solid line shows a bond pointing toward the reader; (iii) the dashed line shows a bond pointing in the opposite direction. Credit: adapted from Wikipedia's Orbital hybridisation article (changed the central letter), licensed under CC BY-SA 3.0.

<figure><img src="image_49.png" /></figure>

FIGURE 1.5 – *Different types of molecular bonds.* We have noted $\vec{p}$, the dipole moment created by the bond. In general, $\vec{p} = \iiint_V \vec{r} \cdot \rho(\vec{r}) dV$, where $\rho(\vec{r})$ is the charge number density. In the case of MgO, it reduces to $e.O.Mg$, where $O.Mg$ is the distance vector between the two ions. The electrons are noted $e^-$, and $\delta^+$ and $\delta^-$ are the excess dipolar charges of both signs. Licensed under CC BY-SA 4.0.

<figure><img src="image_50.png" /></figure>
<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

Covalent bonds are preferentially formed between non metals.

Covalent bonds are of one of the two following types.

**σ bonds** result from the frontal overlap of two s, p or sp<sup>n</sup> (n = 1, 2, 3) orbitals. These bonds have a rotational symmetry around their axis (Fig. I.6.a). The electron density is maximum between the two atoms. It is the strongest covalent bond. The C–C bond of ethane (C<sub>2</sub>H<sub>6</sub>) is a σ bond.

**π bonds** result from the side-by-side overlap of the two lobes of two p orbitals (Fig. I.6.b). The electron density is maximum above and below the plane of the molecule and zero between them. These bonds are weaker than σ bonds. In the double C=C bond of ethylene (C<sub>2</sub>H<sub>4</sub>), there are one σ and one π bonds (Fig. I.6). In the triple C≡C bond of acetylene (C<sub>2</sub>H<sub>2</sub>), there are one σ bond and two π bonds.

Finally, some transitions in interstellar solids involve *antibonding* orbitals. When a bond forms, both a bonding and an antibonding molecular orbitals with different energy levels become available. This is demonstrated for the H<sub>2</sub><sup>+</sup> molecule, with the Schödinger equation, in Chap. 9 of Bransden & Joachain (1983). Bonding orbitals have a lower energy level than the dissociated atoms, thus favoring a stable molecule. On the contrary, the population of an antibonding orbital makes the molecule unstable. For instance, the splitting of molecular orbitals for both σ and π bonds of ethylene (Fig. I.6) are the following.

<figure><img src="image_53.png" /></figure>

We emphasize those are electronic levels of molecules. Molecules also have rotational and vibrational modes that will be discussed in Sect. I.2.2.1.

<figure><img src="image_54.png" /></figure>

FIGURE I.6 – An example of σ and π bonds. We demonstrate the two types of bonds on ethylene (C<sub>2</sub>H<sub>4</sub>). On the left image: the s orbital of each H atom is shown in red; the p orbital of each C atom is shown in green; and the three sp<sup>2</sup> hybridized orbitals of each C atom are in orange. The frontal overlap of the sp<sup>2</sup> hybridized orbitals of two C atoms, as well as the overlap of one sp<sup>2</sup> hybridized C orbital with the s orbital of an H atom, both form a σ bond. On the middle image, we show the plane of the molecule with the p orbital of each C atom in green. The right image shows the overlap of these two p orbitals to form one single π bond, in blue, on both sides of the plane. Credit: adapted from the Chemistry Library, licensed under CC BY-NC 4.0.

I.1.2.2 Ionic Bonds

An *ionic bond* is formed between two atoms of significantly different electronegativities. The electron is transferred from the cation to the anion, resulting in a polar bond. The adhesion is due to

<figure><img src="image_55.png" /></figure>
<figure><img src="image_56.png" /></figure>

long-range Coulomb forces (∝ $1/r^2$) between the two ions. Ionic bonding is non-directional as the electron cloud stays centered around the atoms. The most relevant example to ISD is the $O^{2-}$–$Mg^{2+}$ bond in silicates (Fig. I.5.b).

<figure><img src="image_57.png" /></figure>

Ionic bonds are preferentially formed between a metal and a non metal.

Covalent and ionic bonds are two extreme cases. Most bonds involving at least one non metal are intermediate between both.

### I.1.2.3 Metallic Bonds

Metals can easily be ionized. Bonding several metal atoms therefore results in a lattice of cations bathed in a sea of free valence electrons. The electrons are not attached to a particular atom and can be found anywhere in the solid. This explains the electric and thermal conductivities of metals. Fig. I.5.c represents solid iron.

<figure><img src="image_58.png" /></figure>

Metallic bonds are formed between a large number of metal atoms.

### I.1.2.4 Intermolecular Attraction

Weaker forms of attraction between molecules exist. Their dissociation energy is typically of the order of ≈ 0.1 eV. They are relevant to ISD studies.

**Van der Waals bonds** are due to the induced dipole attraction of neutral atoms and molecules. Their potential energy drops as $1/r^6$. They are in particular responsible for binding graphene sheets together in graphite (Fig. I.5.d).

**Hydrogen bridges** are formed when the induced dipole of the H atom of a molecule is attracted by the induced dipole of a strongly electronegative atom in another molecule. For instance, hydrogen bridges tie the $H_2O$ molecules together in water ice (Fig. I.5.e).

## I.1.3 The Solid State

### I.1.3.1 The Different Types of Solids

There are two main types of solids: *insulators* and *conductors*. Their properties are radically different. Their difference originates in the type of chemical bond making up their crystal lattice.

**Insulators**, also called *dielectrics*, are solids, whose atoms are tied together with covalent or ionic bonds. The valence electrons are therefore located around their specific atoms and can not move freely through the lattice. Consequently, when an electric field is applied, it induces a polarization of the bonds, distorting them, but no current is flowing. For instance, silicates are dielectric materials.

**Conductors** are solids, whose atoms are tied together with metallic bonds. Their valence electrons are therefore free to move through the solid. Consequently, when an electric field is applied, a current is flowing. For instance, iron is a metallic conductor. There are a few non-metal conductors, such as graphite, which is classified as a *semimetal*. This peculiar property is due to the delocalized electrons within the aromatic cycles constituting graphite (cf. Sect. I.1.4).

There is a third, intermediate type of solids, called *semiconductors*. Semiconductors are insulators at T = 0 K and conductors at ambient temperatures. Several cosmic dust candidates belong to this category. We will define it more precisely in Sect. I.1.3.3.

### I.1.3.2 The Band Structure of Solids

A solid can be idealized as a periodic lattice of atoms bonded to each other. The permitted energy levels of a single valence electron, in the periodic electrostatic potential created by this lattice, are

<figure><img src="image_59.png" /></figure>
<figure><img src="image_60.png" /></figure>

a series of continuous functions, also called bands (e.g. Chap. 8 of Ashcroft & Mermin, 1976, for a derivation from the Schrödinger equation). This can be viewed as a generalization of the molecular level splitting (Fig. I.7). The spacing between a large number of levels is so small that it appears continuous. At T = 0 K, the lowest energy bands are filled in priority. Two of these bands are particularly important.

**The valence band** is the highest energy band populated by valence electrons, at T = 0 K.

**The conduction band** is the lowest energy band where electrons can move freely through the solid. It is the band immediately superior to the valence band.

The energy difference between the top of the valence band and the bottom of the conduction band is called the *band gap*, noted E<sub>g</sub> (Fig. I.7).

<figure><img src="image_61.png" /><figcaption>FIGURE I.7 – *Origin of the band structure of a solid.* From the left to the right, we represent: (i) typical discrete atomic levels, (ii) the successive splitting of molecular orbitals, (iii) resulting in the quasi continuous distribution of levels in bands. Electrons are represented with a vertical blue arrow (up or down), corresponding to their spin. Licensed under CC BY-SA 4.0.</figcaption></figure>

### I.1.3.3 The Fermi Level

The probability distribution of identical fermions, such as electrons in a solid, over the energy states of a system at temperature T, is given by the *Fermi-Dirac distribution*:

$$f(E) = \frac{1}{\exp\left(\frac{E - E_F}{kT}\right) + 1}, \qquad (I.2)$$

where *k* is the *Boltzmann constant* (cf. Table B.2), E denotes the different energy levels and E<sub>F</sub> is the *Fermi level*<sup>4</sup>. This distribution is displayed in Fig. I.8.a. The Fermi level is an intrinsic quantity characterizing a solid. It is the energy required to add an electron to the system. It also corresponds to the maximum energy an electron can have at T = 0 K. The latter interpretation of E<sub>F</sub> can be seen in Fig. I.8.a. The blue curve shows Eq. (I.2) at T = 0 K: (i) it gives equal probability to electrons to occupy

<figure><img src="image_62.png" /></figure>

<footnote n="4">4. In the general Fermi-Dirac distribution, the Fermi level, which is proper to solids, is replaced by the *chemical potential* of the system, μ. In our case, the Fermi level is the chemical potential of an electron.</footnote>
<figure><img src="image_63.png" /></figure>

energy levels $E \le E_F$; (ii) it gives zero probability to energy levels $E > E_F$. The actual number density of electrons, $n_e$, is:

$$n_e = \int_{-\infty}^{\infty} g(E) f(E) dE, \qquad (I.3)$$

where $g(E)$ is the density of states per infinitesimal energy bin. This density of states corresponds to the band structure. It is 0 between the bands. We emphasize that $E_F$ can fall between two bands. It does not necessarily correspond to an actual allowed level. This is demonstrated in Fig. I.8.

**Insulators** have their valence and conduction bands widely spread apart (Fig. I.8.b). At ambient temperature, no electron will populate the conduction band. It is another way to see that their valence electrons are localized around their cations.

**Semiconductors** have their valence and conduction bands close to each other (Fig. I.8.c). They are insulators at $T = 0$ K, but their conduction band can be populated at ambient temperature ($E_g$ gets closer to $kT$).

**Conductors** are solids for which the valence and the conduction bands are the same (Fig. I.8.d). The Fermi level is within the band. It is another way to see that the valence electrons are free to move through the lattice at any temperature.

<figure><img src="image_64.png" /><figcaption>FIGURE I.8 – The Fermi level and the different types of solid. The left plot shows the rotated Fermi-Dirac distribution (Eq. I.2), for two values of the temperature, $T = 0$ K and $T \approx 300$ K. The three diagrams on the right show the valence and conduction bands relative to the Fermi level, $E_F$, for insulators, semiconductors and conductors. For conductors, the valence band is also the conduction band. Licensed under CC BY-SA 4.0.</figcaption></figure>

## I.1.4 Interstellar Dust Candidates

We briefly review here the constitution of the most likely ISD grain candidates. Some general properties are given in Table I.3. Their optical properties are extensively discussed in Sect. I.2.2.

### I.1.4.1 Silicates

The different types of silicates are built around silica tetrahedra ($SiO_4^{4-}$), paired with various cations to produce a neutral compound (cf. e.g. Henning, 2010, for a review). The silica tetrahedra have a central $Si^{4+}$ cation tied to four $O^{2-}$ anions with covalent/ionic bonds. In the ISM, the most widely available divalent cations that can be paired with silica tetrahedra are $Mg^{2+}$ and $Fe^{2+}$ (cf. Sect. II.2.3). Silicates have two strong features at 9.7 µm (Si–O stretching) and 18 µm (O–Si–O bending). They are ubiquitous: (i) they are the main constituent of Earth’s crust; (ii) they are also found in Solar system and CircumStellar Dust (CSD); (iii) they account for probably 2/3 of interstellar grain mass

<figure><img src="image_65.png" /></figure>
<figure><img src="image_66.png" /></figure>

(e.g. Draine, 2003a); (iv) their features are observed in distant galaxies, in absorption (e.g. Marcillac et al., 2006) and in emission (e.g. Hony et al., 2011). Interstellar silicates are widely amorphous (e.g. Kemper et al., 2004). Crystalline silicates have additional distinctive narrow features, due to SiO₄ as well as (Fe,Mg)–O vibrations, in the 9.0–12.5 µm and 14-22 µm ranges, with a few bands above 33 µm. The following two types of silicates are the most relevant to ISD (cf. Table I.3).

**Olivine.** Olivine have the general formula $(Mg,Fe)_2SiO_4$, with different proportions of Mg and Fe. Its crystalline structure is represented on Fig. I.9.b. The two following compounds are the extreme cases of the Fe-to-Mg ratio.

**Forsterite** is the Mg-end of the series, with formula $Mg_2SiO_4$.

**Fayalite** is the Fe-end of the series, with formula $Fe_2SiO_4$.

Olivine have an olive green color (Fig. I.10.a).

**Pyroxene.** Pyroxene have the general formula $(Mg,Fe)SiO_3$, with different proportions of Mg and Fe. They are constituted of silica tetrahedron chains, sharing one O atom (Fig. I.9.c), which explains their stoichiometry. The two following compounds are the extreme cases of the Fe-to-Mg ratio.

**Enstatite** is the Mg-end of the series, with formula $MgSiO_3$.

**Ferrosilite** is the Fe-end of the series, with formula $FeSiO_3$.

Pyroxene can be darker than olivine (Fig. I.10.b).

<figure><img src="image_67.png" /></figure>

In general, silicates are translucent minerals. They are gemstones, used in jewelry.

### I.1.4.2 Hydrogenated Amorphous Carbon

This is a broad class of solids, noted a-C(:H) (a notation introduced by Jones, 2012b). Carbon atoms can be paired in the following ways (Fig. I.9.d).

**Aromatic cycles** are hexagonal rings made of six sp² hybridized C atoms. Two of the three available sp² orbitals of each C atom make σ bonds, tying the cycle together. The last sp² orbital can be used to make a σ bond with another C atom, extending the compound, or with an H atom, ending the solid in this direction. The six remaining p orbitals of the cycle make a sort of ring-shaped π bond. The electrons of these bonds are delocalized, they do not belong to a specific C atom, but they are confined to the cycle. This is why compounds with aromatic cycles have some properties of metals: electric conductivity and shiny appearance.

**Aliphatic groups** are centered around a sp³ hybridized C atom. Its four sp³ orbitals can be paired to other C atoms or to H atoms, forming σ bonds.

**Olefinic bonds** are alkene-type double bonds between two sp² hybridized C atoms (Fig. I.6). There is one σ bond bridging two sp² orbitals, and one π bond linking the p orbital of each C atom.

The hydrogenation of a-C(:H) influences directly their band gap (Jones, 2012b). Generally, H-poor a-C(:H), which can be noted a-C, are sp² dominated (aromatic/olefinic), and have a low band gap ($E_g \approx 0.4–0.7$ eV). On the contrary, H-rich a-C(:H), which can be noted a-C:H, are mostly aliphatic (sp³), and have a larger band gap ($E_g \approx 1.2–2.5$ eV). The aromatic domains are responsible for bright features at 3.3, 6.2, 7.7, 8.6 and 11.3 µm, that will be extensively discussed in Sect. III.2.1.1, whereas the main aliphatic feature is at 3.4 µm. An important feature at 2175 Å (Sect. II.2.1) is thought to originate in the transition between the π and π* bands of sp² domains (e.g. Draine & Li, 2001).

<figure><img src="image_68.png" /></figure>

a-C(:H) tend to be more opaque than silicates (Fig. I.10.c).

<figure><img src="image_69.png" /></figure>
<figure><img src="image_70.png" /></figure>

FIGURE 1.9 – *Structure of interstellar dust candidates.* The (a) image shows the silica tetrahedron, which is the building block of silicates. The (b) image shows the structure of crystalline olivine. Some silica tetrahedra are pointing toward the reader above the yellow plane, and some are pointing backward. The latter appear dimmer and the Si atom are visible in red. The (c) image shows the structure of crystalline pyroxene. Its most important feature is the chain of alternate silica tetrahedra. They share one O atom, explaining why the stoichiometry of pyroxene is different from olivine. The (d) image shows the diversity of carbon pairing in an a-C(:H). Every C atom is sp² hybridized, except the aliphatic C, which is sp³ hybridized. Every bond is a σ bond except: (i) the 6 delocalized electrons within each aromatic cycles are π bond pairing of p orbitals; (ii) one of the olefinic bonds is a π bond. The example we have shown corresponds to a very small grain. In a larger a-C(:H), most of these bonds would be linked into the rest of a contiguous 3D network (e.g. Micelotta et al., 2012, for more realistic structures). The (e) image shows that graphite is the stack of graphene sheets. Each graphene sheet is made exclusively of aromatic cycles. The (f) image displays a few different PAHs. Licensed under CC BY-SA 4.0.

<figure><img src="image_71.png" /></figure>

FIGURE 1.10 – *Appearance of various minerals.* The (a) and (b) images show crystalline silicates. The soot powder (c) is an approximate analog of a-C(:H), on the a-C end, except that soot can contain radicals with O, N, etc. These images are useful to visualize the differences in optical properties of these compounds. They however show macroscopic samples. Even the powder in image (c) is made of particles much larger than 1 µm, which is the maximum size of interstellar grains. Credit: (a) forsterite from Rob LAVINSKY, licensed under CC BY-SA 3.0; (b) enstatite from Rob LAVINSKY, licensed under CC BY-SA 3.0; (c) soot from Wikipedia, not licensed; (d) graphite from Rob LAVINSKY, licensed under CC BY-SA 3.0; (e) PAHs from the Astrochemistry Lab, NASA Ames Research Center, with permission from Lou ALLAMANDOLA.
<figure><img src="image_72.png" /></figure>

### I.1.4.3 Graphite

Graphite is a mineral made of the stacking of graphene sheets, bonded by Van Der Waals interactions (Sect. I.1.2). Graphene sheets are planar compounds exclusively constituted of aromatic cycles (Fig. I.9.e). Pure graphite is solely made of sp² carbon. Its aromaticity explains its shiny silver metallic appearance (Fig. I.10.d). It has a strong $π → π^*$ transition around 2175 Å. The exact central wavelength however depends on the size and shape of the particles, and pure graphite seems too wide to account for the interstellar feature (e.g. Draine & Malhotra, 1993; Voshchinnikov, 2004; Papoular & Papoular, 2009). Graphite also has a broad band at 30 µm, seen parallel to the sheets, which corresponds to the oscillation frequency of the delocalized π electrons (e.g. Venghaus, 1977; Draine & Li, 2007).

### I.1.4.4 Polycyclic Aromatic Hydrocarbons (PAH)

PAHs are a class of molecules made of aromatic cycles, with peripheral H atoms (Fig. I.9.f). They have the aromatic features of a-C, as well as the $π → π^*$ transition around 2175 Å (e.g. Joblin et al., 1992). Similarly to graphite, the exact central wavelength depends on the particle size and shape (e.g. Duley & Seahra, 1998). They can be colorful (cf. Fig. I.10.e). They are highly flammable and carcinogenic.

<table><caption>TABLE I.3 – General properties of interstellar dust candidates. The values of the density and melting temperature are approximate. They vary between samples and experimental conditions.</caption><thead><tr><th>Name</th><th>Stoichiometry</th><th>Density</th><th>Melting</th><th>Main spectroscopic features</th></tr></thead><tbody><tr><td colspan="5" style="text-align: center; font-weight: bold;">SILICATES</td></tr><tr><td>Forsterite</td><td>Mg<sub>2</sub>SiO<sub>4</sub></td><td>3.3 g/cm<sup>3</sup></td><td>2200 K</td><td>9.7, 18 µm</td></tr><tr><td>Fayalite</td><td>Fe<sub>2</sub>SiO<sub>4</sub></td><td>4.4 g/cm<sup>3</sup></td><td>1500 K</td><td>9.7, 18 µm</td></tr><tr><td>Enstatite</td><td>MgSiO<sub>3</sub></td><td>3.2 g/cm<sup>3</sup></td><td>2100 K</td><td>9.7, 18 µm</td></tr><tr><td>Ferrosilite</td><td>FeSiO<sub>3</sub></td><td>4.0 g/cm<sup>3</sup></td><td>1200 K</td><td>9.7, 18 µm</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">CARBONACEOUS</td></tr><tr><td>a-C(:H)</td><td>C<sub>n</sub>H<sub>m</sub></td><td>1.8-2.1 g/cm<sup>3</sup></td><td>N/A</td><td>2175 Å, 3.3, 3.4, 6.2, 7.7, 8.6, 11.3 µm</td></tr><tr><td>Graphite</td><td>C<sub>n</sub></td><td>2.3 g/cm<sup>3</sup></td><td>3600 K</td><td>2175 Å, 30 µm</td></tr><tr><td>PAH</td><td>C<sub>n</sub>H<sub>m</sub></td><td>2.2 g/cm<sup>3</sup></td><td>N/A</td><td>2175 Å, 3.3, 6.2, 7.7, 8.6, 11.3 µm</td></tr></tbody></table>

## I.2 The Interaction of Light with Solids

The interaction of an electromagnetic wave with dust grains results in the three following phenomena (cf. Fig. I.11).

**Absorption:** a fraction of the electromagnetic energy is stored into the grain;

**Scattering:** the wave vector of the fraction that is not absorbed changes direction, its field polarization changes, but its frequency is not affected;

**Emission:** the energy stored in the grain is ulteriorly re-emitted in the IR.

The sum of absorption and scattering is called extinction. These three phenomena can be modeled, assuming valence electrons are harmonic oscillators. This way, the response to an electromagnetic wave of bonds in a dielectric or free electrons in a metal can be quantified. This is illustrated in Fig. I.12. Throughout this manuscript, we use $λ$, $ν = c/λ$ and $ω = 2πν$ to respectively denote the wavelength, frequency and angular frequency of an electromagnetic wave, c being the speed of light (cf. Table B.2).

<figure><img src="image_73.png" /></figure>
<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

**FIGURE I.11 – Absorption, scattering and emission.** The images on the right are the Horsehead nebula. Licensed under CC BY-SA 4.0. Credit: Horsehead nebula images from NASA, ESA, and the Hubble Heritage Team (AURA/STScI); ESO, licensed under CC BY 4.0.

<figure><img src="image_76.png" /></figure>

**FIGURE I.12 – Effect of an electromagnetic wave on a dielectric.** An incoming, circularly polarized, electromagnetic wave is figured in magenta. The cube on the right represents a solid. The nuclei, assumed to be fixed, are the red spheres. The valence electrons are the green ellipsoids. They are displaced out of their equilibrium positions by the electromagnetic wave, inducing a time-dependent polarization. Licensed under CC BY-SA 4.0.

<figure><img src="image_77.png" /></figure>
<figure><img src="image_78.png" /></figure>

## I.2.1 Bonds as Harmonic Oscillators

The harmonic oscillator model is particularly useful to describe the way bonds react to electromagnetic waves.

### I.2.1.1 The Harmonic Oscillator Amplitude

The position along the x-axis of a unidimensional harmonic oscillator of mass, *m*, as a function of time, *t*, follows the equation:

$$ m \underbrace{\frac{d^2 x(t)}{dt^2}}_{\text{inertia}} + b \underbrace{\frac{dx(t)}{dt}}_{\text{friction}} + \underbrace{k_e x(t)}_{\text{restoring force}} = \underbrace{F(t)}_{\text{external force}}, \qquad (I.4) $$

where F is the external force applied to the oscillator, $k_e$ is the strength of the restoring force, and b is a dissipation constant. This equation is simply the Newton law ($F = m\ddot{x}$), where the force, $F$, has three components: (i) the external force, F, displacing the oscillator out of its equilibrium position ($x = 0$); (ii) the restoring force, $-k_e x$, which is proportional to the distance, meaning it is stronger when the oscillator is farther away from its equilibrium position; (iii) the friction force, $-b\dot{x}$, proportional to the velocity, having the effect of slowing down the oscillator.

In the case of the motion of an electron, excited by the Lorentz force of a complex, harmonic plane electromagnetic wave with angular frequency $\omega$, $F(t) = eE_0 \exp(-i\omega t)$, Eq. (I.4) can be rewritten:

$$ \frac{d^2 x(t)}{dt^2} + \gamma \frac{dx(t)}{dt} + \omega_0^2 x(t) = \frac{e E_0 \exp(-i \omega t)}{m_e}, \qquad (I.5) $$

where $m_e$ is the electron mass. We have also introduced the *natural frequency*, $\omega_0$, and the *damping constant*, $\gamma$:

$$ \omega_0 \equiv \sqrt{\frac{k_e}{m_e}} \quad \text{and} \quad \gamma \equiv \frac{b}{m_e}. \qquad (I.6) $$

In this case, the restoring force is created by the atom's electrostatic potential well, and the friction can be interpreted as collisions of the electron with the lattice. The solution to Eq. (I.5) has the form $x(t) = x_0 \exp(-i\omega t)$, with complex amplitude (e.g. Levi, 2016):

$$ x_0 = \frac{e E_0}{m_e (\omega_0^2 - \omega^2 - i \omega \gamma)}. \qquad (I.7) $$

It is important to consider both $\vec{x}$ and $\vec{E}$ as complex quantities, since there is a phase shift induced by the dissipation term. The module of $x_0$, giving the physical value of the amplitude, is:

$$ |x_0| = \frac{e|E_0|}{m_e \sqrt{(\omega^2 - \omega_0^2)^2 + \gamma^2 \omega^2}}. \qquad (I.8) $$

This is the classic harmonic oscillator solution. It is represented in Fig. I.13. The amplitude is maximum at the *resonant frequency*, $\omega_r = \sqrt{\omega_0^2 - \gamma^2/2}$. If there is no dissipation ($\gamma = 0$), the amplitude becomes infinite, and the electron escapes.

### I.2.1.2 The Plasma Frequency

Free electrons oscillate around heavy cations at the *plasma frequency*, $\omega_p$. Its formula is (cf. e.g. Chap. 1 of Krügel, 2003, for a simple proof):

$$ \omega_p \equiv \sqrt{\frac{n_e e^2}{m_e \epsilon_0}}, \qquad (I.9) $$

<figure><img src="image_79.png" /></figure>
<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /><figcaption>FIGURE I.13 – Amplitude of a forced harmonic oscillator. The left image represents the effect of an electromagnetic field, $\vec{E}$, on an orbital. The displacement between the nucleus and the most probable position of the electron induces a permanent dipole, $\vec{p} = -e \cdot \vec{x}$. In this simple picture, the nucleus (cation) is assumed fixed and the electron oscillates around it. The right panel shows the amplitude of the motion of the electron (Eq. I.8) as a function of frequency, for different values of the friction, $\gamma$. We have normalized both axes in order to display dimensionless quantities. Licensed under CC BY-SA 4.0.</figcaption></figure>

where $n_e$ is the number density of free electrons. It applies to metals, as well as actual gaseous plasmas. These media absorb and scatter electromagnetic waves with frequencies lower than $\omega_p$. For instance, in the case of the Earth's ionosphere, $n_e \approx 10^{12} \text{ m}^{-3}$, thus $\omega_p \approx 60$ MHz. This explains why amateur radio operators communicate over long distances at frequencies lower than this value, to benefit from the reflection of their transmission on the ionosphere (e.g. Perry et al., 2018). In the case of a metal, with density $n_e \approx 10^{29} \text{ m}^{-3}$, $\omega_p \approx 20$ PHz, corresponding to a wavelength $\lambda_p \approx 0.1 \text{ µm}$, in the UltraViolet (UV; cf. Table A.4). This explains the shiny appearance of metals, as they are able to reflect the visible light, which has a lower frequency than UV photons. It happens that the expression of $\omega_p$ also appears in the optical properties of dielectrics, and we will use it extensively.

### I.2.1.3 The Dielectric Function

In a dielectric, an electromagnetic wave polarizes the bonds. If we consider each bond as a dipole with moment $\vec{p}$, the polarization density is defined as:

$$ \vec{P} = n_e \vec{p}, \qquad (I.10) $$

where $n_e$ is the number density of valence electrons. The induced polarization density is directly related to the electric field:

$$ \vec{P} = \epsilon_0 \chi \vec{E}, \qquad (I.11) $$

where $\chi$ is the electric susceptibility. The electric displacement field, $\vec{D}$, which accounts for the charge displacement induced by an electric field $\vec{E}$ is defined as:

$$ \vec{D} = \epsilon \vec{E}, \qquad (I.12) $$

where $\epsilon$ is the electric permittivity of the medium. The relative electric permittivity, $\epsilon_r$ is defined such that: $\epsilon = \epsilon_r \epsilon_0$. It is a macroscopic quantity, as no medium is truly continuous. At atomic scales, Eq.

<figure><img src="image_82.png" /></figure>
<figure><img src="image_83.png" /></figure>

(I.12) can be broken into two terms:

$$ \vec{D} = \underbrace{\epsilon_0 \vec{E}}_{\text{vacuum between atoms}} + \underbrace{\vec{P}}_{\text{induced dipoles}} = \epsilon_0 \underbrace{(1 + \chi)}_{\epsilon_r} \vec{E}. \quad (I.13) $$

The second equality derives from Eq. (I.11). It also implies that $\epsilon_r = 1 + \chi$. The induced polarization is, using Eq. (I.7) and Eq. (I.10):

$$ \vec{P} = n_e e \vec{x} = \underbrace{\frac{n_e e^2}{m_e \omega_0^2 - \omega^2 - i\omega\gamma}}_{\epsilon_0\chi} \vec{E}. \quad (I.14) $$

Finally, a bit of algebra and introducing the plasma frequency (Eq. I.9), gives:

$$ \epsilon_r(\omega) = 1 + \underbrace{\frac{\omega_p^2(\omega_0^2 - \omega^2)}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}}_{\epsilon_1(\omega)} + i \underbrace{\frac{\omega_p^2\gamma\omega}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}}_{\epsilon_2(\omega)}. \quad (I.15) $$

This dispersion relation is called the *dielectric function*. It is displayed in Fig. I.14.a. It can be related to the refractive index of the material, *m*. Indeed, plane electromagnetic waves propagating in a dielectric have a phase velocity $v_{ph} = 1/\sqrt{\epsilon\mu}$, where $\mu$ is the *magnetic susceptibility* (cf. e.g. Chap. 7 of Jackson, 1999, for a derivation from Maxwell's equations). This phase velocity can also be expressed as a function of the speed of light, *c*: $v_{ph} = c/m$, where *m* is the refractive index. Since we can decompose $\mu = \mu_r\mu_0$, and because $\epsilon_0\mu_0 = 1/c^2$, we have:

$$ m(\omega) = \sqrt{\epsilon_r(\omega)\mu_r(\omega)} = n(\omega) + ik(\omega). \quad (I.16) $$

The refractive index is sometimes referred to as the *optical constants*, or simply the “n and k”. In a nonmagnetic medium, $\mu_r = 1$, thus:

$$ \begin{cases} \epsilon_1(\omega) = n^2(\omega) - k^2(\omega) \\ \epsilon_2(\omega) = 2n(\omega)k(\omega). \end{cases} \quad (I.17) $$

The two complex quantities $\epsilon(\omega)$ and $m(\omega)$ contain the same information. Eq. (I.15) corresponds to one single type of harmonic oscillator, that is to one mode of one type of bond. An actual dielectric is usually the linear combination of several resonances, such as Eq. (I.15), with different sets of $\omega_p$, $\omega_0$, and $\gamma$.

### I.2.1.4 Harmonic Oscillator Cross-Section

The flux carried by a plane electromagnetic wave is given by the time average of the *Poynting vector*:

$$ \langle |\vec{\mathcal{P}}| \rangle = \frac{\epsilon_0 c}{2} |\vec{E}_0|^2. \quad (I.18) $$

The power radiated by a dipole, harmonically oscillating, is (cf. e.g. Chap. 9 of Jackson, 1999):

$$ W_{\text{rad}} = \frac{2}{3} \frac{\langle |\ddot{\vec{p}}|^2 \rangle}{4\pi\epsilon_0 c^3} = \frac{\omega^4 e^2 |x_0|^2}{12\pi\epsilon_0 c^3}. \quad (I.19) $$

This power is the radiation in response to the excitation by the incident wave. This is the scattering contribution. The *scattering cross-section* of this harmonic oscillator is thus simply: $C_{\text{sca}} = W_{\text{rad}} / \langle |\vec{\mathcal{P}}| \rangle$. Replacing $|x_0|$ by Eq. (I.8), we obtain:

$$ C_{\text{sca}}(\omega) = C_{\text{T}} \frac{\omega^4}{(\omega^2 - \omega_0^2)^2 + \gamma^2\omega^2}, \quad (I.20) $$

<figure><img src="image_84.png" /></figure>
<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /><figcaption>FIGURE I.14 – Idealized optical constants. Both panels show dielectric functions where the bonds (in the dielectric case) or the free electrons (in the case of a conductor) are treated as harmonic oscillators. Panel (a) displays Eq. (I.15). We see that the imaginary part, $\epsilon_2$, which corresponds to the absorption, peaks around the resonant frequency and drops rapidly to zero on both sides. Panel (b) displays Eq. (I.28). We see that the absorption by a conductor rises rapidly below the plasma frequency. It is another way to witness the fact that metals can absorb any light below $\omega_p$. At high frequency, $\epsilon_2 \to 0$. Licensed under CC BY-SA 4.0.</figcaption></figure>

where we have introduced the *Thomson cross-section*:

$$C_T = \frac{8\pi}{3} \left( \frac{e^2}{4\pi\epsilon_0 m_e c^2} \right)^2 \approx 6.66 \times 10^{-29} \, \text{m}^2. \qquad (I.21)$$

Now, the absorbed power comes from the dissipation into the solid. The dissipation force in Eq. (I.5) is $\vec{F}_{\text{dis}} = -m_e \vec{\gamma} \vec{x}$. The dissipated power is thus the work of this force:

$$W_{\text{dis}} = m_e \gamma |\vec{x}|^2 = \frac{1}{2} m_e \gamma \omega^2 |x_0|^2. \qquad (I.22)$$

The absorption cross-section of the harmonic oscillator is therefore: $C_{\text{abs}} = W_{\text{dis}} / \langle |\vec{\mathcal{P}}| \rangle$. Using Eq. (I.8) and Eq. (I.21), we obtain:

$$C_{\text{abs}}(\omega) = \frac{e^2}{m_e \epsilon_0 c} \frac{\gamma \omega^2}{(\omega^2 + \omega_0^2)^2 + \gamma^2 \omega^2} \qquad (I.23)$$

It is interesting to look at the limiting behavior of both $C_{\text{sca}}$ and $C_{\text{abs}}$ (see also Chap. 1 of Krügel, 2003). At high frequency ($\omega \gg \omega_0$; short wavelength), we have:

$$\left\{ \begin{array}{l} C_{\text{sca}}(\omega) \approx C_T \\ C_{\text{abs}}(\omega) \approx \frac{e^2}{4m_e\epsilon_0 c} \frac{\gamma}{\omega^2}. \end{array} \right. \qquad (I.24)$$

Around the resonant frequency ($\omega \approx \omega_0$), we have:

$$\left\{ \begin{array}{l} C_{\text{sca}}(\omega) \approx \frac{C_T}{4} \frac{\omega_0^2}{(\omega - \omega_0)^2 + (\gamma/2)^2} \\ C_{\text{abs}}(\omega) \approx \frac{e^2}{4m_e \epsilon_0 c} \frac{\gamma}{(\omega - \omega_0)^2 + (\gamma/2)^2}. \end{array} \right. \qquad (I.25)$$

<figure><img src="image_87.png" /></figure>
<figure><img src="image_88.png" /></figure>

It shows that around the resonant frequency, both C_sca and C_abs have Lorentz profiles centered at ω₀, with Full Width at Half Maximum (FWHM), γ.

**At short frequency** (ω ≪ ω₀; long wavelength):

$$ egin{cases} C_{\text{sca}}(\omega) \approx C_T \left(\frac{\omega}{\omega_0}\right)^4 \\ C_{\text{abs}}(\omega) \approx \frac{e^2 \gamma}{m_e \epsilon_0 c \omega_0^2} \left(\frac{\omega}{\omega_0}\right)^2. \end{cases} \qquad (\text{I.26}) $$

Those approximations are particularly useful.

For dielectrics, C_sca ∝ 1/λ⁴ and C_abs ∝ 1/λ² at long wavelength.

### I.2.1.5 Optical Constants of Conductors

Eq. (I.15) is valid for a dielectric, as it assumes the medium is only constituted of dipoles. This is not the case in a conductor where there are also free charges. An electromagnetic wave induces a current, ⃗j, related to the electric field, ⃗E, by the conductivity, σ:

$$ \vec{j} = \sigma \vec{E} = n_e e \vec{v}, \qquad (\text{I.27}) $$

where the second equality relates the current to its microscopic origin, the velocity of free electrons, ⃗v. Maxwell's equations for plane waves give (e.g. Chap. 7 of Jackson, 1999):

$$ \epsilon_r(\omega) = \underbrace{\epsilon_d(\omega)}_{\text{bound electrons}} + \underbrace{i \frac{\sigma(\omega)}{\epsilon_0 \omega}}_{\substack{\text{free electrons}}} , \qquad (\text{I.28}) $$

where ε_d(ω) is the leftover dielectric term. We can apply the harmonic oscillator model again to these free electrons. The difference is that there is no restoring force, ω₀ = 0. From Eq. (I.7), the velocity of free electrons becomes:

$$ |\vec{v}| = \frac{e|\vec{E}|}{m_e(\gamma - i\omega)}. \qquad (\text{I.29}) $$

Injecting this quantity in Eq. (I.27), we obtain:

$$ \sigma(\omega) = \epsilon_0 \omega_p^2 \frac{\gamma}{\gamma^2 + \omega^2} + i \epsilon_0 \omega_p^2 \frac{\omega}{\gamma^2 + \omega^2}. \qquad (\text{I.30}) $$

Focussing on the free electron term in Eq. (I.28), i.e. assuming ε_d = 0, we get:

$$ \epsilon_r(\omega) = 1 - \underbrace{\frac{\omega_p^2}{\gamma^2 + \omega^2}}_{\epsilon_1(\omega)} + i \underbrace{\frac{\gamma \omega_p^2}{\omega \gamma^2 + \omega^2}}_{\epsilon_2(\omega)}. \qquad (\text{I.31}) $$

It is also known as the *Drude model*. This equation is displayed in Fig. I.15.b. Interestingly enough, the cross-sections of Eq. (I.20) and Eq. (I.23) apply also to conductors, with ω₀ = 0. We can easily derive their limiting behavior.

Conductors have the same behavior than dielectrics at long wavelength: C_sca ∝ 1/λ⁴ and C_abs ∝ 1/λ².

<figure><img src="image_89.png" /></figure>
<figure><img src="image_90.png" /></figure>

### I.2.1.6 The Kramers-Kronig Relations

The residue theorem implies that, if $f(x)$ is a complex function of the complex variable $x$, analytical over $\Im(x) \ge 0$, and dropping faster than $1/|x|$, we have the relation:

$$f(\omega) = \frac{1}{i\pi} P \int_{-\infty}^{\infty} \frac{f(x)}{x-\omega} dx, \qquad (I.32)$$

with $\omega$ real and positive. We have used the *Cauchy principal value*:

$$P \int_{-\infty}^{\infty} \frac{f(x)}{x-\omega} dx = \lim_{\delta \to 0} \left( \int_{-\infty}^{\omega-\delta} \frac{f(x)}{x-\omega} dx + \int_{\omega+\delta}^{\infty} \frac{f(x)}{x-\omega} dx \right), \qquad (I.33)$$

which is simply an integral avoiding the singularity in $x = \omega$. Decomposing $f(x) = f_1(x) + if_2(x)$, we obtain cross-relations between $f_1(x)$ and $f_2(x)$. These general mathematical relations are usually applied to the susceptibility, from which we derive the dielectric function (e.g. Chap. 21 of Draine, 2011):

$$\left\{ \begin{aligned} \epsilon_1(\omega) - 1 &= \frac{2}{\pi} P \int_0^\infty \frac{x\epsilon_2(x)}{x^2 - \omega^2} dx \\ \epsilon_2(\omega) &= -\frac{2}{\pi} \omega P \int_0^\infty \frac{\epsilon_1(x) - 1}{x^2 - \omega^2} dx. \end{aligned} \right. \qquad (I.34)$$

These relations are known as the *Kramers-Kronig relations*.

**Implications.** We only need to specify $\epsilon_1$ or $\epsilon_2$ at all frequencies, and can use Eq. (I.34) to derive the other one. These relations are used to check laboratory data consistency (e.g. Zubko et al., 1996).

**Interpretation.** They are a consequence of the causality requirement for a linear system (here, we have $\vec{P} = \epsilon_0\chi\vec{E}$). In our case, they impose that the response of the polarization does not precede the effect of the electric field. Sect. 62 of Landau & Lifshitz (1960), Chap. 21 of Draine (2011), and Chap. 2 of Krügel (2003) discuss these relations more extensively.

**Constraint on the Cross-Section.** They give some constraints on the long wavelength behavior of the dielectric function. Let's assume that $\epsilon_2(\omega) \propto \omega^{\beta-1}$ for $\omega < \delta$, for an arbitrary $\delta$. The first relation tells us that:

$$\epsilon_1(0) = 1 + \frac{2}{\pi} \int_0^\delta x^{\beta-2} dx + \int_\delta^\infty \frac{\epsilon_2(x)}{x} dx. \qquad (I.35)$$

The second integral is finite by requirement. For the first integral to be finite, we need to have $\beta > 1$. At long wavelength, Eq. (I.15) tells us that, for a dielectric:

$$\left\{ \begin{aligned} \epsilon_1(\omega) &\xrightarrow{\omega \to 0} 1 + \left(\frac{\omega_p}{\omega_0}\right)^2 &= \text{const} \\ \epsilon_2(\omega) &\xrightarrow{\omega \to 0} \frac{\omega_p^2 \gamma}{\omega_0^4} \omega \ll \epsilon_1(\omega). \end{aligned} \right. \qquad (I.36)$$

We will see in Eq. (I.46) that, at long wavelength, $C_{\text{abs}} \propto \epsilon_2 / [(\epsilon_1 + 2)^2 + \epsilon_2^2] / \lambda$. Using Eq. (I.36), we get $C_{\text{abs}}(\lambda) \propto \epsilon_2(\lambda) / \lambda \propto \lambda^{-\beta}$. We will see in Sect. III.1.2 that this $\beta$ parameter is sometimes referred to as the *emissivity index*.

Assuming that $C_{\text{abs}}(\lambda) \propto \lambda^{-\beta}$ at long wavelengths, we need to have $\beta > 1$.

<figure><img src="image_91.png" /></figure>
<figure><img src="image_92.png" /></figure>

<figure><img src="image_93.png" /><figcaption>FIGURE I.15 – *Molecular transitions*. The different types of transitions are illustrated with the CO molecule. Licensed under CC BY-SA 4.0.</figcaption></figure>

## I.2.2 Grain Optical Properties

### I.2.2.1 Why Are Most Dust Features in the MIR?

All but one spectral features of the interstellar grain candidates we have listed in Table I.3 are in the MIR. This is a general trend (e.g. Table 1 of van der Tak et al., 2018a). It can be understood by making an analogy with the different types of molecular transitions (cf. e.g. Chap. 2 of Tielens, 2005, for a review). Those are illustrated in Fig. I.15.

**Electronic transitions** are transitions between the quantum harmonic oscillator levels of the bonding electron. In the case of solids, they are transitions between bands, such as the $\pi \rightarrow \pi^*$ transition of aromatic carbon, at 2175 Å (cf. Sect. I.1.4). The natural frequency of these resonances is given by Eq. (I.6): $\omega_0 = \sqrt{k_e/m_e}$. The energy of these resonances is comparable to the binding energy, or the band gap. They typically range between $\approx 4$ and $\approx 20$ eV. They are thus in the UV domain ($\lambda \approx 0.06 - 0.30$ µm).

**Vibrational transitions** are associated with the stretching or bending of a bond. These modes involve the motion of the nuclei, which are much heavier than the electrons. Their frequency is $\omega_v = \sqrt{k_v/\mu_{1,2}}$, where $\mu_{1,2} = m_1m_2/(m_1 + m_2)$ is the reduced mass of the two atoms, $m_1$ and $m_2$. Typically, $\mu_{1,2} = 0.9, 6, 10 \times m_p$ for C–H, C–C and Si–O bonds, respectively ($m_p$ is the proton mass; cf. Table B.2). At first order, the new force constant is similar to previously, $k_e \approx k_v$. The frequency is now reduced by a factor $\approx \sqrt{m_e/\mu_{1,2}} \approx 0.007 - 0.02$. These transitions are thus in the MIR ($\lambda \approx 2 - 40$ µm). They are the most relevant transitions for ISD.

**Rotational transitions** are associated with the rotation of the molecule. Their energy depends on the centrifugal force, which reduces the frequency by a factor $\approx m_e/m_p$. These transitions are thus in the millimeter regime. Most dust grains do not have detectable rotational transitions, because of their inertia. Only the smallest, charged grains have a non-negligible rotational emission that will be discussed in Sect. II.2.2.3.

### I.2.2.2 Dielectric Functions of Realistic Materials

The dielectric functions of Eq. (I.15) and Eq. (I.28) correspond to simple cases where there is only one type of oscillator. Realistic materials have more complex structures, with several modes per bond. Deriving dielectric functions of potential interstellar grain analogs is the subject of a rich literature. There are three types of approaches to determine the dispersion relation of a medium.

**The theoretical knowledge** of the microscopic structure of the crystal can be used to determine the different resonances that we have demonstrated in Sect. I.2.1. The resonant or plasma fre-

<figure><img src="image_94.png" /></figure>
<figure><img src="image_95.png" /></figure>

quency, as well as the collisional rates (γ) have to be known. The Kramers-Kronig relations (Sect. I.2.1.6) can be used to obtain complementary constraints.

**Laboratory data** about the spectral profile of some features, or the density of the material, or any relevant characteristics can also be used. The ISM community is active in this field, as our results largely depend on the accurate values of atomic and molecular data. Several teams focus their research effort on laboratory measures of dust analogs.

**Astronomical observations** can be used to constrain some features. The most famous example is the use of observations of the 9.7 and 18 µm band profiles to build the optical properties of astronomical silicates, by Draine & Lee (1984). Indeed, we have seen in Sect. I.1.4 that there is a diversity of silicate composition, and we do not know which one is relevant to ISD (it is likely a mixture).

These approaches are not exclusive and are usually combined as observations and laboratory data are always partial. The work by Draine & Lee (1984) was the first study to use these principles to derive the UV-to-mm dielectric functions of graphite and astronomical silicates. We now have a better knowledge of the dispersion relations of several important materials: (i) silicates (e.g. Laor & Draine, 1993; Weingartner & Draine, 2001a; Draine, 2003b,c); (ii) amorphous carbon (e.g. Rouleau & Martin, 1991; Zubko et al., 1996; Jones, 2012a,b,c); (iii) graphite (e.g. Laor & Draine, 1993; Draine, 2003b,c, 2016); (iv) PAHs (e.g. Li & Draine, 2001; Draine & Li, 2007); and (v) composite grains (e.g. Köhler et al., 2014, 2015). Fig. I.16 shows a few examples.

### I.2.2.3 Computing Grain Cross-Sections

The dielectric functions are intensive quantities characterizing the bulk optical properties of solids, but independent of their size and shape. To compute usable absorption and scattering cross-sections, there is one last step to do. Let's assume our grains are spheres of radius a.

* The extinction cross-section of the grain can be written:

$$C_{\text{ext}}(\lambda, a) = \underbrace{\pi a^2}_{\text{geometric cross-section}} \times \underbrace{Q_{\text{ext}}(\lambda, a)}_{\text{efficiency}}. \qquad (\text{I.37})$$

This expression is simply the geometric cross-section, $\pi a^2$, times an extinction efficiency, $Q_{\text{ext}}$, which is a dimensionless quantity. We can express the scattering and absorption cross-section the same way:

$$C_{\text{sca}}(\lambda, a) = \pi a^2 Q_{\text{sca}}(\lambda, a) \qquad (\text{I.38})$$

$$C_{\text{abs}}(\lambda, a) = \pi a^2 Q_{\text{abs}}(\lambda, a). \qquad (\text{I.39})$$

Out of these three efficiencies, only two are independent, as we have $Q_{\text{ext}} = Q_{\text{sca}} + Q_{\text{abs}}$.

* Another useful quantity that can be derived from the efficiencies is the albedo, quantifying the fraction of the incident light that is scattered by the grain:

$$\tilde{\omega}(\lambda, a) = \frac{C_{\text{sca}}(\lambda, a)}{C_{\text{ext}}(\lambda, a)} = \frac{Q_{\text{sca}}(\lambda, a)}{Q_{\text{ext}}(\lambda, a)}. \qquad (\text{I.40})$$

* Let's call θ the angle between the directions of the incident and scattered light. The probability distribution of scattering angles is called the scattering phase function:

$$\Phi(\cos\theta, \lambda, a) = \frac{1}{C_{\text{sca}}(\lambda, a)} \frac{d\mathcal{C}_{\text{sca}}(\cos\theta, \lambda, a)}{d\Omega}, \qquad (\text{I.41})$$

where $\mathcal{C}_{\text{sca}}(\cos\theta, \lambda, a)$ is the differential scattering cross-section (i.e. the cross-section for scattering in a given direction), and $d\Omega = d\cos\theta d\phi$ is the solid angle element. It is normalized over all directions, such that:

$$\iint_{\Omega} \Phi(\cos\theta, \lambda, a) d\Omega = 2\pi \int_{-1}^{1} \Phi(\cos\theta, \lambda, a) d\cos\theta = 1. \qquad (\text{I.42})$$

<figure><img src="image_96.png" /></figure>
<figure><img src="image_97.png" /></figure>

<figure><img src="image_98.png" /></figure>

FIGURE I.16 – *Dielectric functions of different materials.* In each panel, we show the real part of the dielectric function as $\epsilon_1 - 1$ (blue) and $1 - \epsilon_1$ (green), as this quantity can be negative. The imaginary part, $\epsilon_2$, is in red. Panel (a) shows the astronomical silicates by Draine (2003b,c). Panel (b) shows the ACAR type of amorphous carbon by Zubko et al. (1996). These are produced by arc discharges between amorphous carbon electrodes in a 10 mbar Ar atmosphere. Panel (c) and (d) show the graphite by Draine (2003b,c). Graphite is anisotropic, as we have seen in Fig. I.9.e. It is usually approximated by mixing 2/3 of the optical properties parallel to the graphene sheets (c), and 1/3 perpendicular (d). Licensed under CC BY-SA 4.0.
For isotropic scattering, we have $\Phi(\cos\theta, \lambda, a) = 1/4\pi$. The first moment of this distribution is called the asymmetry parameter, defined as:

$$g(\lambda, a) = \langle \cos\theta \rangle = 2\pi \int_{-1}^{1} \Phi(\cos\theta, \lambda, a) \cos\theta d\cos\theta. \qquad (I.43)$$

This parameter is a direct product of Mie theory. Forward and backward scattering correspond to $\langle\cos\theta\rangle \approx 1$ and $\langle\cos\theta\rangle \approx -1$, respectively, whereas isotropic scatterers have $\langle\cos\theta\rangle \approx 0$. There are approximate analytical phase functions. The most famous is from Henyey & Greenstein (1941):

$$\Phi(\cos\theta, \lambda, a) = \frac{1}{4\pi (1 + g^2(\lambda, a) - 2g(\lambda, a) \cos\theta)^{3/2}}. \qquad (I.44)$$

Other distributions have been proposed (e.g. Draine, 2003b).

The treatment to compute $Q_{sca}$, $Q_{abs}$ and $g$ depends on the value of the size parameter:

$$x = \frac{2\pi a}{\lambda}. \qquad (I.45)$$

As long as we do not zoom in scales where the hypothesis of a continuous medium breaks down, that is scales of a few Å (i.e. grains made of a few atoms, or hard X-ray photons), the estimation of the efficiencies of a grain only depends on $x$ and $m$. The Mie theory (cf. e.g. Chap. 4 of Bohren & Huffman, 1983)<fnref n="5" /> is the central tool to compute grain cross-sections. It is a numerical method, exactly solving Maxwell's equations for the scattering of a plane electromagnetic wave by a homogeneous sphere of known refractive index, $m(\lambda)$. Several regimes can be identified, depending on the value of the size parameter. They are illustrated in Fig. I.18. In Fig. I.19, we show the actual cross-sections of silicate and graphite grains of different sizes.

**Geometrical optics.** It is the regime for which $x \gg 1$ (cf. e.g. Chap. 7 of Bohren & Huffman, 1983). For interstellar grains, which have submicronic sizes, it corresponds to UV wavelengths and shorter. This regime is more relevant to circumstellar dust, where grains can be significantly larger. In geometrical optics, the undulatory nature of light is put aside. Instead, light is modeled as rays, using the formalism of Fresnel. Mie theory is valid in this regime, but numerical problems start arising.

* The important feature of this regime is that, $Q_{sca} \approx Q_{abs} \approx 1$ (cf. Fig. I.18.b). This can also be seen in panels (a) to (d) of Fig. I.19. When the size of the grain increases, the range where both $Q_{sca}$ and $Q_{abs}$ are flat extends to shorter wavelengths. The cross-sections are independent of wavelength, as the grains are effectively behaving as opaque circular screens. It also means that the cross-section is proportional to the area of the grain, but independent of its volume.

* $Q_{sca} \approx Q_{abs} \approx 1$ implies that $Q_{ext} \approx 2$, meaning that the extinction cross-section is twice the geometric cross-section. This counter-intuitive result is called the extinction paradox (Chap. 4 of Bohren & Huffman, 1983, for an extensive discussion). It is a real diffraction effect, that is not actually predicted by the methods of geometrical optics, for which $Q_{ext} \approx 1$. Its resolution lies in accounting for the diffraction around the grain. It is illustrated in Fig. I.17. The exact interpretation of this paradox is still debated nowadays (e.g. Berg et al., 2011).

* Another feature of this regime is that grains are efficient forward scatterers ($\langle\cos\theta\rangle \approx 1$; Fig. I.18.c).

**The Mie regime.** It corresponds to grain sizes comparable to the wavelength of the incident light ($x \approx 1$). Mie theory is valid outside this regime, but this is the regime where none of the other approximations are valid.

<footnote n="5">5. See also B. Draine's public code, `bhmie.f`, implementing the algorithm in Appendix A of Bohren & Huffman (1983).</footnote>
<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

**FIGURE I.17** – *Illustration of the extinction paradox.* We have represented two situations. In panel (a), we show the radiation from a distant star passing through a circular pupil of radius *a*, and projected onto the plane of a detector. We assume that: (i) the star is very distant, so that the incident wave can be considered planar; (ii) the detector is at a distant from the pupil much larger than *a*. The observed pattern is a series of Airy rings. In panel (b), we show the same stellar radiation absorbed and scattered by a dust grain of the same radius *a* as the pupil, at the same distance from a detector. The observed pattern is made of the same Airy rings, but there is a bright spot in the center. This property is a result of Babinet's theorem. This theorem is based on the fact that the wave function of a plane wave, $\psi(x, y)$, diffracted by a pupil of transmission $T(x, y)$, is proportional to the Fourier transform of the pupil, $\mathcal{F}(T)$. Here, the direction represented by the blue arrow is the *z* axis. The *x* and *y* axes are the coordinates in the plane of the pupil and of the detector, both being parallel. If we have two pupils, $T_{(a)}$ and $T_{(b)}$, that are complementary, that is $T_{(a)}(x, y) + T_{(b)}(x, y) = 1, \forall (x, y)$, we have, by linearity of the Fourier transform $\psi_{(a)}(x, y) + \psi_{(b)}(x, y) \propto \mathcal{F}(T_{(a)}) + \mathcal{F}(T_{(b)}) = \delta(x, y)$, where $\delta$ is the Dirac distribution centered in the middle of the screen. In our case, the two complementary pupils are the circular hole in panel (a), and the grain, appearing as a circular screen to the stellar radiation, in panel (b). The intensity is the squared module of the wave function, $I(x, y) = |\psi(x, y)|^2$. We see that $\psi_{(b)}(x, y) = \psi_0\delta(x, y) - \psi_{(a)}(x, y)$, thus $I_{(b)}(x, y) = I_{\text{spot}}\delta(x, y) + I_{(a)}(x, y)$, confirming that the pattern in panel (b) is the pattern in panel (a) plus a bright spot. In panel (b), the bright spot is just the incident wave (rays parallel to the *z* axis). We note that $I_{(a)}(x, y) = I_{(b)}(x, y)$, except in (0, 0). The incident light of intensity $I_0$ on the grain surface ($\pi a^2$) is absorbed. Thus, the absorbed power is $P_{\text{abs}} = \pi a^2 I_0$. Now, the Airy pattern is the fraction diffracted by the grain contour, that is the scattered light. Babinet's theorem tells us that it is identical in both cases we have represented. We clearly see in panel (a) that the scattered power is also $P_{\text{sca}} = \pi a^2 I_0$. The grain thus extinct a power $P_{\text{ext}} = 2\pi a^2 I_0$. This is the extinction paradox: (i) the flux incident on the grain surface is fully absorbed; (ii) the contour of the grain scatters the same power at small angles, corresponding to the Airy rings in both panels. Licensed under CC BY-SA 4.0.

* The optical properties have non-trivial features, depending on *x* and on the actual resonances of *m*. For instance, panels (a) and (c) of Fig. I.19 show the 9.7 and 18 µm features of silicates. These features disappear only for radii $a \gtrsim 10$ µm, as the geometrical optics regime extends in the mid-IR (MIR; cf. Table A.4), in this case.

* The scattering pattern is non-trivial, $\langle \cos\theta \rangle$ varying between 0 and 1. It is schematically represented in the central cartoon at the top of Fig. I.18, using the Eq. (I.44). It shows that backward scattering is likely, although forward scattering is more probable.

* Finally, we notice small oscillations in the visible and near-IR (NIR; cf. Table A.4) ranges, mostly visible in the $Q_{\text{abs}}$ of micronic size grains (panels (c) and (d) of Fig. I.19). These patterns are called *interference structure*. They are due to the interference between the incident and scattered waves. They are more prominent for weakly absorbent material, such as silicate, which is more transparent than graphite (cf. Sect. I.1.4).

**The Rayleigh regime.** It corresponds to grain sizes significantly smaller than the wavelength of the incident light, $x \ll 1$ (cf. e.g. Chap. 5 of Bohren & Huffman, 1983). In the case of interstellar grains,

<figure><img src="image_101.png" /></figure>
<figure><img src="image_102.png" /></figure>

<figure><img src="image_103.png" /><figcaption>FIGURE 1.18 – *Mie, Rayleigh and geometric optics regime.* The top pictures represent the scattering pattern (in magenta) of a spherical grain (red sphere), in the three regimes. The direction of the incident photon is shown in yellow. The scattering pattern has been calculated using the Henyey & Greenstein (1941) phase function. The length of each wiggly arrow is proportional to its scattering probability. The three panels plot below show the optical properties of a *a* = 0.1 µm radius silicate, by Draine (2003b,c). Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_104.png" /></figure>
<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /><figcaption>FIGURE I.19 – *Cross-sections of silicate and graphite grains.* The color lines show the optical properties of grains with different radius, *a*, from Draine (2003b,c). Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_107.png" /></figure>

it applies essentially to the NIR regime and longward. The refraction index needs to be small, too: $|m|x \ll 1$. The solutions are analytic (e.g. Li, 2008):

$$ \left\{ \begin{aligned} C_{\text{sca}}(\lambda, a) &= \frac{128\pi^5}{3} \left| \frac{\epsilon_r(\lambda) - 1}{\epsilon_r(\lambda) + 2} \right|^2 \frac{a^6}{\lambda^4} \\ C_{\text{abs}}(\lambda, a) &= 24\pi^2 \frac{\epsilon_2(\lambda)}{(\epsilon_1(\lambda) + 2)^2 + \epsilon_2^2(\lambda)} \frac{a^3}{\lambda}. \end{aligned} \right. \qquad (I.46) $$

* Eq. (I.46) and panels (a) to (d) of Fig. I.19 show that $Q_{\text{sca}}/a^4$ and $Q_{\text{abs}}/a$ are independent of radius in this regime. This explains why small grains have a negligible albedo, whereas large grains are efficient scatterers.

* The fact that $Q_{\text{abs}}/a$ is independent of radius implies that the absorption cross-section is proportional to the grain volume: $C_{\text{abs}} \propto a^3$. The dust mass can thus be probed by measures in absorption or in emission. An interpretation of this property is that, the wavelength being comparable to the size of the grain, each bond can interact with the electromagnetic field, independently of its location within the solid. The interaction of the incident light with the grain is thus proportional to the total number of oscillators, which is proportional to the mass.

* It explains why scattering appears negligible in this regime: $(C_{\text{sca}} \propto a^6) \ll (C_{\text{abs}} \propto a^3)$. Finally, we can see that scattering is isotropic ($\langle \cos \theta \rangle \approx 0$).

The absorption cross-section of most interstellar grains, in the NIR-to-mm window, is proportional to their volume. The dust mass can thus be probed by absorption or emission measures.

### I.2.2.4 Beyond Homogeneous Spheres

Mie theory is restricted to homogeneous spheres. However, there are several observational indications that this hypothesis is not fully accurate (cf. Sect. IV.2.1.1).

* Grains are rapidly formed and destroyed in the ISM, implying that they likely are composites of several materials with different dielectric functions, and voids.

* The polarization of starlight and of ISM emission in the far-IR (FIR; cf. Table A.4) indicates that at least some of the grains are elongated.

There are several methods to estimate cross-sections of grains beyond the hypothesis of homogeneous spheres.

**Effective medium theory (EMT).** EMT is a class of methods to replace the individual dielectric functions of a composite material by an average, $\epsilon_{av}(\lambda)$. Cross-sections can then be estimated using Mie theory or any other approximation. EMT assumes that the different domains are smaller than the wavelength and well-mixed in the grain. There are different mixing rules (cf. e.g. Chap. 8 of Bohren & Huffman, 1983). It seems that their accuracy depends on the type of sample they are applied to, as independent studies find better agreements with one or the other (e.g. Abeles & Gittleman, 1976; Perrin & Lamy, 1990).

**Maxwell Garnett's rule** assumes that the medium is constituted of a matrix with dielectric function $\epsilon_m(\lambda)$, and some inclusions. The inclusions can be of N different types, with dielectric functions $\epsilon_i(\lambda)$ ($i = 1...N$), and volume filling factors, $\phi_i$. It is implicit that $\phi_i \ll 1$.

$$ \epsilon_{\text{av}}(\lambda) = \frac{\epsilon_m(\lambda) + \sum_{i=1}^{N} \phi_i c_i \epsilon_i(\lambda)}{1 + \sum_{i=1}^{N} \phi_i c_i}, \qquad (I.47) $$

with:

$$ c_i = \frac{3\epsilon_m(\lambda)}{\epsilon_i(\lambda) + 2\epsilon_m(\lambda)}. \qquad (I.48) $$

<figure><img src="image_108.png" /></figure>
<figure><img src="image_109.png" /></figure>

**Bruggeman's rule** does not put a hierarchy between a predominant matrix and a few inclusions. The dielectric function is the solution of:

$$ \sum_{i=1}^{N} \phi_i \frac{\epsilon_i(\lambda) - \epsilon_{av}(\lambda)}{\epsilon_i(\lambda) + 2\epsilon_{av}(\lambda)} = 0. \qquad (I.49) $$

**Ellipsoids in the Rayleigh regime.** They have general analytical solutions (e.g. Chap. 5 of Bohren & Huffman, 1983). When the three axes of the ellipsoid are aligned on the coordinates x, y, z and if the electric field is along x, we have:

$$ \left\{ \begin{aligned} C_{\text{sca}}(\lambda, a) &= \frac{128\pi^5}{27} \left| \frac{\epsilon_r(\lambda) - 1}{1 + (\epsilon_r(\lambda) - 1)L_x} \right|^2 \frac{a^6}{\lambda^4} \\ C_{\text{abs}}(\lambda, a) &= \frac{8\pi^2}{3} \Im \left[ \frac{\epsilon_r(\lambda) - 1}{1 + (\epsilon_r(\lambda) - 1)L_x} \right] \frac{a^3}{\lambda}, \end{aligned} \right. \qquad (I.50) $$

where $L_x$ is the *shape factor*. Noting $l_a > l_b$ the two lengths, oblate spheroids have dimensions along the three axes ($l_a, l_b, l_x$), whereas prolate spheroids have dimensions ($l_a, l_b, l_x$) (cf. Fig. I.20). With these notations, the shape factor is (e.g. Chap. 22 of Draine, 2011):

$$ \left\{ \begin{aligned} L_x &= \frac{1 + \xi^2}{\xi^2} \left[ 1 - \frac{\arctan \xi}{\xi} \right] & \text{for oblate spheroids} \\ L_x &= \frac{1 - \xi^2}{\xi^2} \left[ \frac{1}{2\xi} \ln \left( \frac{1 + \xi}{1 - \xi} \right) - 1 \right] & \text{for prolate spheroids,} \end{aligned} \right. \qquad (I.51) $$

with $\xi^2 = |1 - (l_b/l_a)^2|$. In case of randomly oriented ellipsoids, we simply need to take the arithmetic mean of the cross-sections along the three axes.

<figure><img src="image_110.png" /><figcaption>FIGURE I.20 – Oblate and prolate spheroids. Credit: volumes produced with Mathematica.</figcaption></figure>

**Discrete Dipole Approximation (DDA).** DDA (Purcell & Pennypacker, 1973)<fnref n="6" /> allows the user to model complex composite grains as arbitrary arrays of independent domains. These domains are approximated by a series of discrete dipoles, which must be much smaller than the incoming wavelength. This method is computer intensive, but very flexible (cf. e.g. the results of Köhler et al., 2015; Ysard et al., 2018). We show some of the results of Köhler et al. (2015) in Fig. I.21. This figure exhibits an important result we will discuss later:

⊗ the addition of mantles and the aggregation of grains tend to increase the absorptivity per unit mass in the FIR window.

<figure><img src="image_111.png" /></figure>

<footnote n="6">6. See the public code DDSCAT by Draine & Flatau (1994).</footnote>
<figure><img src="image_112.png" /><figcaption>FIGURE 1.21 – Absorption cross-sections of grain aggregates computed with DDA. The curves in the left panel are the absorption cross-sections of grains of radius $a = 0.2 \ \mu\text{m}$: $C_{\text{abs}}(a, \lambda)/m(a) = 3/(4\rho) \times Q_{\text{abs}}(a, \lambda)/a$, where $\rho$ is the density. The four curves correspond to the four main mixtures of the THEMIS model (Jones et al., 2013, 2017), represented in the right panel: (i) the Core-Mantle (CM) mixture is made amorphous Forsterite and Enstatite with Fe and FeS (troilite) inclusions, and aliphatic amorphous carbon (a-C:H); both grains are coated with an aromatic mantle (a-C); (ii) the Core-Mantle-Mantle (CMM) is the CM mixture with additional coating by aliphatic material; (iii) the Aggregate-Mantle-Mantle (AMM) mixture is constituted of aggregated CMM grains; (iv) the Aggregate-Mantle-Mantle-Ice (AMMI) mixture is the AMM aggregates coated with water ice. The CM optical properties are from Jones et al. (2013). The CMM, AMM and AMMI optical properties have estimated by Köhler et al. (2015) using DDA. Notice the apparition of the 3 µm water ice band (Fig. II.12) in the AMMI model. Licensed under CC BY-SA 4.0.</figcaption></figure>

### I.2.2.5 Polarization

The real part of the electric field of a monochromatic, plane electromagnetic wave, propagating along the z-axis, can be written at time $t$ and at $z = 0$:

$$ \Re(\vec{E}) = \vec{E}_1 \cos \omega t + \vec{E}_2 \sin \omega t, \qquad (I.52) $$

where $\vec{E}_1 \perp \vec{E}_2$ (cf. e.g. Chap. 2 of Bohren & Huffman, 1983). This is the parametric equation of an ellipse in the $(x, y)$ plane (cf. Fig. I.22.a). It is the most general case, called elliptical polarization. It is fully characterized by the modules of $\vec{E}_1$ and $\vec{E}_2$, the angle $\varphi$ and the rotation direction $\eta = \pm 1$ (-1: clockwise; +1:counterclockwise). Particular cases are: (i) linear polarization, if $|\vec{E}_1| = 0$ or $|\vec{E}_2| = 0$; and (ii) circular polarization, if $|\vec{E}_1| = |\vec{E}_2|$.

**The Stokes parameters.** They constitute a four element vector easier to observationally measure than the ellipse parameters. They are noted $(I, Q, U, V) = \vec{S}$. I is the total intensity of the beam, that can be partially polarized. The three others can be expressed as:

$$ \left\{ \begin{aligned} Q &= (|\vec{E}_1|^2 - |\vec{E}_2|^2) \cos(2\varphi) \\ U &= (|\vec{E}_1|^2 - |\vec{E}_2|^2) \sin(2\varphi) \\ V &= 2\eta|\vec{E}_1||\vec{E}_2|. \end{aligned} \right. \qquad (I.53) $$

<figure><img src="image_113.png" /></figure>
<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /><figcaption>FIGURE I.22 – Stokes parameters. Panel (a) displays the notations we have used to parametrize the el- liptical polarization. Panels (b) to (g) demonstrate the type of polarization each parameter represents (linear or circular, in different directions), when the rest is zero. Licensed under CC BY-SA 4.0.</figcaption></figure>

The interpretation of these parameters is illustrated in Fig. I.22. They allow us to decompose the light into its linear and circular polarization components. The intensity of the polarized light is $I_p = \sqrt{Q^2 + U^2 + V^2} \le I$. It is also frequent to quote the *linearly polarized intensity*:

$$P = \sqrt{Q^2 + U^2}, \qquad (I.54)$$

and the *linear polarization fraction*, P/I.

**Grain alignment.** Asymmetric grains tend to be aligned with the magnetic field. If we consider the simple case of spheroidal grains (cf. Fig. I.20): (i) the rotation axis of oblate grains is along their symmetry axis; (ii) the rotation axis of prolate grains is perpendicular to their long axis, their cross-section thus needs to be integrated over their spinning dynamics (e.g. Guillet et al., 2018). The rotation axis of the grains tends to align with the magnetic field, $\vec{B}$. This is represented in panels (b) and (c) of Fig. I.23. Several mechanisms have been proposed to explain this alignment (cf. Andersson et al., 2015, for a review). Nowadays, *Radiative Alignment Torques* (RAT; Dolginov & Mitrofanov, 1976) are favored, because they provide the best account of the observational constraints. This complex scenario is based on the fact that irregular grains have different cross-sections for clockwise and counterclockwise circularly polarized light. Light scattering on such grains therefore provides a torque that increases the angular momentum of the grains. If these grains have paramagnetic inclusions (such as iron), the grain rotation precesses and aligns with $\vec{B}$. Although the alignment is caused by the radiation field, it is independent of its direction. This mechanism becomes inefficient at high optical depth, which is consistent with observations.

**Polarization by scattering.** It is represented on Fig. I.23.a. When an incident beam is scattered by a grain (this grain does not have to be asymmetric), the electric field component in the scattering plane is diminished, inducing a polarization perpendicular to the plane. The larger the scattering angle is, the larger the polarization. The polarization of an incident Stokes vector, $\vec{S}_i$, resulting in a scattered beam, $\vec{S}_s$, is described as: $\vec{S}_s = \vec{M}\vec{S}_i$, where $\vec{M}$ is the 4 × 4 *Müller matrix* (cf. Chap. 3 of Bohren & Huffman, 1983, for different examples of Müller matrices). This polarization process is not related to grain alignment with the magnetic field, but it depends on the distribution of stars and dust clouds (e.g. Wood, 1997).

<figure><img src="image_116.png" /></figure>
**Dichroic extinction.** It is the selective extinction of the electric field oscillating along the major axis of an elongated grains. It is represented in Fig. I.23.b. Since grains in the diffuse ISM tend to align their rotation axis with the magnetic field, they polarize starlight parallel to $\vec{B}$. For this reason, the polarization of starlight has historically been used to map the magnetic field of the MW (Mathewson & Ford, 1970).

**Polarized emission.** The polarization of the emission by elongated grains has been predicted by Stein (1966). Such grains emit IR light preferentially polarized along the direction of their major axis. Since their major axis is perpendicular to the magnetic field, their IR emission is perpendicular to $\vec{B}$. It is represented in Fig. I.23.c. The FIR polarized emission has been extensively used to map the magnetic field in the MW, with the Planck satellite (e.g. Planck Collaboration et al., 2016d).

Grain polarization is parallel to the magnetic field in the visible and perpendicular in the IR: $\vec{P}_{vis} \parallel \vec{B}$ and $\vec{P}_{IR} \perp \vec{B}$.

## I.2.3 Heat Capacities

In Sect. I.2.1, presenting the harmonic oscillations of valence electrons, we argued that the energy dissipation was due to collisions with the lattice. The energy absorbed by the grain is thus redistributed throughout the lattice and stored in the harmonic oscillations of its atoms.

### I.2.3.1 Distribution of Harmonic Oscillators

The energy levels of a one-dimensional quantum harmonic oscillator, of natural frequency $v$, are (e.g. Chap. 2 of Atkins & Friedman, 2005):

$$E_n = h\nu \left(n + \frac{1}{2}\right) \quad \text{with } n = 0, 1, \dots, \qquad (I.55)$$

where $h$ is the *Planck constant* (cf. Table B.2). At thermal equilibrium, the probability distribution of a large ensemble of such harmonic oscillators, at temperature $T$, is the *Boltzmann distribution*:

$$p_n(T) = \frac{\exp\left(\frac{-E_n}{kT}\right)}{Z(T)}, \qquad (I.56)$$

where we have introduced the *partition function*:

$$Z(T) = \sum_{n=0}^{\infty} \exp\left(-\frac{E_n}{kT}\right) = \frac{\exp\left(-\frac{h\nu}{kT}\right)}{1 - \exp\left(-\frac{h\nu}{kT}\right)}. \qquad (I.57)$$

The second equality comes from injecting Eq. (I.55) into the first equality. The mean energy of this ensemble of oscillators is the first moment of the distribution in Eq. (I.56):

$$\langle E \rangle = \frac{h\nu}{2} + \frac{h\nu}{\exp\left(\frac{h\nu}{kT}\right) - 1}. \qquad (I.58)$$

Since each one of the N atoms of the lattice has three degrees of freedom (along the three Cartesian axes $x, y, z$), the number of oscillators is $3N^7$. The *internal energy* of the grain, which is the energy

<footnote n="7">7. This is for $N \gg 1$. The actual number of degrees of freedom is $3N-6$, subtracting the three translatory and three rotational possible motions of the grain as a whole.</footnote>
<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /><figcaption>FIGURE I.23 – The three types of grain-induced polarization of light. Panel (a) represents the unpolarized starlight radiation being scattered by a grain. The electric field of the scattered light oscillates predominantly perpendicularly to the scattering plane. The larger the scattering angle is, the higher the linear polarization fraction is. Panels (b) and (c) represent a single prolate grain whose rotation axis is aligned with the magnetic field, $\vec{B}$. The case of oblate grains is more trivial as their rotation axis is their symmetry axis. Panel (b) illustrates dichroic extinction of starlight in the visible/near-IR. The grain extinct preferentially the component of the electric field oscillating in its rotation plane. The extincted starlight is thus polarized parallel to $\vec{B}$. Panel (c) shows the IR emission of the same prolate grain is polarized along its major axis, perpendicular to $\vec{B}$. In both panels (b) and (c), we have represented the optimal case, when the magnetic field is perpendicular to the sightline. In the more general case, the polarization will be relative to the projection of $\vec{B}$ on the plane of the sky. Licensed under CC BY-SA 4.0.</figcaption></figure>

stored in all its oscillators, is thus:

$$U(T) = 3N\langle E \rangle = 3N\hbar v \left( \frac{1}{2} + \frac{1}{\exp\left(\frac{\hbar v}{kT}\right) - 1} \right). \qquad (I.59)$$

<figure><img src="image_119.png" /></figure>
<figure><img src="image_120.png" /></figure>

### I.2.3.2 Debye's Model

Eq. (I.59) considers that each atom oscillates independently of its neighbors. In reality, there are collective vibrational modes in a crystal lattice. These modes actually are sound waves, propagating at the sound speed of the material, $c_s$. Because the size of a grain is finite, the number of these possible modes is quantified. Indeed, if L is the size of the grain along one dimension, the wavelength of the modes along this dimension is $\lambda_n = 2L/n$, with $n = 1, 2, \dots$ (cf. Fig. I.24). The shortest possible wavelength corresponds to oscillations of adjacent atoms in opposition of phase: $\lambda_D = 2d_{at}$, where $d_{at}$ is the interatomic distance. These quantified modes can be treated as quasi-particles, called *phonons*. These phonons have energies $hc_s/\lambda_n$, thus:

$$E_n = \frac{nhc_s}{2L} \quad \text{for } n = 1, \dots, n_D. \qquad (I.60)$$

We now need to integrate Eq. (I.58) over the different modes:

$$U(T) = U_0 + \int_0^{v_D} \frac{hv}{\exp\left(\frac{hv}{kT}\right) - 1} g(v) \, dv, \qquad (I.61)$$

where $g(v)$ is the density of modes with frequency $v$, and $U_0$ is a constant coming from the 1/2 term in Eq. (I.58). It can be shown that (cf. Chap. III.E of Diu et al., 1997):

$$g(v) = \frac{9Nv^2}{v_D^3}, \qquad (I.62)$$

where the Debye frequency can be explicit as a function of the density of atoms, $n_{at}$:

$$v_D = c_s \sqrt[3]{\frac{9n_{at}}{4\pi}}. \qquad (I.63)$$

From $v_D$, we can also define the *Debye temperature*, $T_D \equiv h\nu_D/k$. Eq. (I.61) thus becomes:

$$U(T) = U_0 + \frac{9N}{v_D^3} \int_0^{v_D} \frac{hv^3}{\exp\left(\frac{hv}{kT}\right) - 1} dv, \qquad (I.64)$$

and the *Debye heat capacity* can be derived:

$$C(T) = \frac{\partial U}{\partial T} = 9kN \left(\frac{T}{T_D}\right)^3 \int_0^{T_D/T} \frac{x^4 e^x}{(e^x - 1)^2} dx. \qquad (I.65)$$

It is represented in Fig. I.25.a.

**The Dulong-Petit regime** is the limiting behavior of Eq. (I.65) at high temperature, namely:

$$C(T) \approx 3Nk \quad \text{for } T \gg T_D. \qquad (I.66)$$

It is the classical expression of heat capacity. It is constant becomes it assumes that energy can be indifferently stored in oscillators, ignoring their limited number.

**The Debye regime** is the low-temperature limit of Eq. (I.65), namely:

$$C(T) \approx \frac{12\pi^4}{5} Nk \left(\frac{T}{T_D}\right)^3 \quad \text{for } T \ll T_D. \qquad (I.67)$$

It accounts for the quantification of the modes. It provides a correct agreement with laboratory measurements.

<figure><img src="image_121.png" /></figure>
<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /><figcaption>FIGURE I.24 – *Phonon modes.* We represent the simplest case of a string of atoms (red spheres). The total length of the solid is materialized by the yellow horizontal line. The two atoms at each end of this line are fixed. The modes are thus quantified. The shortest possible wavelength is $2d_{at}$, corresponding to the $n = n_D$ mode. Licensed under CC BY-SA 4.0.</figcaption></figure>

### I.2.3.3 Heat Capacities of Realistic Materials

The Debye model is an idealization providing a good approximation. It has however several limitations.

**Conduction electrons** contribute to the heat capacity of metals, and dominate at low temperatures. Their contribution to the heat capacity is (cf. e.g. Chap. 2 of Ashcroft & Mermin, 1976):

$$C_{\text{cond}}(T) = \frac{\pi^2 N k T}{2 T_F}, \qquad (I.68)$$

where $T_F = E_F/k$ is the Fermi temperature (Eq. I.2).

**Laboratory data** can be used to determine the Debye and Fermi temperatures of the compound. If the structure of the grain is too complex, the heat capacity can be fitted on experimental measurements (e.g. Draine & Li, 2001). We show the heat capacity of various interstellar grain candidates in Fig. I.25.b.

## I.2.4 Heating and Cooling

### I.2.4.1 Kirchhoff's Law

Let's consider a grain at thermal equilibrium with a radiation source, such as the light from a star. The *specific intensity* received by the grain, $I_v(\lambda, \Omega)$, is the electromagnetic power per unit frequency, area (A) and solid angle ($\Omega$) $^8$: $dE_\star = I_v dt d\nu d\Omega$ (we discuss this quantity in more details in Sect. III.1.1). The *absorption coefficient* of this grain, $\alpha(\lambda)$, is the fraction of this specific intensity it absorbs per unit length, $l$: $dI_v = -\alpha I_v dl$. The *emission coefficient* of this grain, $j_v(\lambda)$, is the power it emits per unit frequency, volume (V) and solid angle (it is isotropic): $dE_{em} = j_v dt d\nu dV d\Omega$. Kirchhoff (1860)'s law states that the ratio $j_v(\lambda)/\alpha(\lambda) = f_v(T, \lambda)$ is a universal function depending only on T and $\lambda$ (e.g. Robitaille, 2009, for a review). Planck (1900) later gave an analytical expression of this empirical function, assuming the energy levels were discrete, providing a quantum formulation of the *black body* radiation. It became the *Planck function*, $f_v(T, \lambda) = B_v(T, \lambda)$, where:

$$B_v(T, \lambda) = \frac{2hc}{\lambda^3} \frac{1}{\exp\left(\frac{hc}{\lambda k T}\right) - 1} \qquad (I.69)$$

<figure><img src="image_124.png" /></figure>

<footnote n="8">8. Throughout this manuscript, we use the subscript v to exclusively denote *spectral densities*, that is quantities per unit frequency, $f_v$. Such a quantity can also be expressed per unit wavelength: $f_\lambda = f_v d\nu/d\lambda = f_v c/\lambda^2$. Quantities depending on the frequency, but not per unit frequency, should not be written with subscript v: $\kappa \bar{\nu} \rightarrow \kappa(\nu)$.</footnote>
<figure><img src="image_125.png" /><figcaption>FIGURE I.25 – Heat capacities. Panel (a) shows the Debye model (Eq. (I.65); blue), with the two limiting regimes: (i) the classical Dulong-Petit regime (magenta dashed line); and (ii) the quantum Debye regime (red dashed line). Panel (b) shows the heat capacities of realistic materials: PAH, graphite and silicate from Draine & Li (2001) and a-C(:H) from Jones et al. (2013). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Opacity.** The mass absorption coefficient of a grain, $\kappa_{\text{abs}}(\lambda)$, is its absorption cross-section per unit mass: $\kappa_{\text{abs}}(\lambda) = \alpha(\lambda)/\rho$. For a single spherical grain, it is, using Eq. (I.39):

$$ \kappa_{\text{abs}}(a, \lambda) = \underbrace{\frac{C_{\text{abs}}(a, \lambda)}{\frac{4\pi}{3} a^3 \rho}}_{\text{mass}} = \frac{3}{4\rho} \frac{Q_{\text{abs}}(a, \lambda)}{a}. \qquad (I.70) $$

This quantity is often referred to as the *opacity*. In this manuscript, we however extend this term to its scattering component, too. We will therefore call $\kappa \equiv \kappa_{\text{ext}} = \kappa_{\text{sca}} + \kappa_{\text{abs}}$, the *opacity*, $\kappa_{\text{abs}}$ and $\kappa_{\text{sca}}$ being called the *absorption* and *scattering opacities*, respectively. We have seen in Sect. I.2.2 that $Q_{\text{abs}}/a$ is practically independent of radius for most interstellar grains in the NIR regime and longward, thus:

☞ the NIR-to-mm opacity of interstellar grains having the same homogeneous composition is independent of their radius.

**Emissivity.** The emissivity of a grain, $\epsilon_v(\lambda)$, is the power it emits per unit frequency and mass ($dm = \rho dV$): $dE_{\text{em}} = \epsilon_v(\lambda)\rho dt dv d\Omega/4\pi$. The last differential element simply denotes an average over solid angles. We thus see that $\epsilon_v = 4\pi j_v/\rho$. Kirchhoff's law then becomes:

$$ \epsilon_v(\lambda) = 4\pi\kappa_{\text{abs}}(\lambda) \times B_v(T, \lambda). \qquad (I.71) $$

Eq. (I.71) is the emission spectrum of grains at thermal equilibrium with the radiation field. We show a few examples in Fig. I.26. The limiting behavior of Eq. (I.71) when $\lambda \gg hc/kT$ is given by the Rayleigh-Jeans law:

$$ \epsilon_v(\lambda) \approx 8\pi kT \frac{\kappa_{\text{abs}}(\lambda)}{\lambda^2}. \qquad (I.72) $$
<figure><img src="image_126.png" /></figure>

<figure><img src="image_127.png" /><figcaption>FIGURE I.26 – Emissivity of grains at thermal equilibrium with the radiation field. We show the emission spectrum of spherical graphite (a) and silicates (b) from Draine (2003b,c), at different equilibrium temperatures (Eq. I.71). We have overlaid in green the Rayleigh-Jeans approximation ($\beta = 2$ for both compounds). Licensed under CC BY-SA 4.0.</figcaption></figure>

**Modified Black Body (MBB).** A MBB is an idealized body, at thermal equilibrium with the radiation field, that does not perfectly absorb all frequencies. In other words, it has a non zero albedo. It is an imperfect black body, sometimes also called *grey body*. Eq. (I.71) is a MBB. In the ISM literature, MBB usually refers to the case where we approximate the opacity as a power-law:

$$ \kappa_{\text{abs}}(\lambda) \approx \kappa_0 \left(\frac{\lambda_0}{\lambda}\right)^{\beta}, \qquad (\text{I.73}) $$

where $\lambda_0$ is a reference wavelength, and $\kappa_0$, the opacity at $\lambda_0$. This approximation was popularized by Hildebrand (1983). We have seen in Sect. I.2.1 that $\beta \approx 2$ for typical grains, and that we must have $\beta > 1$ in order to satisfy the Kramers-Kronig relations (cf. Sect. I.2.1.6). Eq. (I.72) implies that $\epsilon_v(\lambda) \propto \lambda^{2+\beta}$, in the Rayleigh-Jeans regime. We have shown this relation in Fig. I.26.

**The net flux radiated by a black body is:**

$$ F_{\text{BB}}(T) = \int_{0}^{\infty} \pi B_{\nu}(T, \nu) d\nu = \sigma T^{4}, \qquad (\text{I.74}) $$

where $\sigma$ is the *Stefan-Boltzmann constant* (cf. Table B.2). In the case of a MBB, the emitted power per unit mass is:

$$ \frac{P_{\text{MBB}}(\beta, T)}{M_{\text{MBB}}} = \int_{0}^{\infty} 4\pi\kappa_{0} \left(\frac{\lambda_{0}}{\lambda}\right)^{\beta} B_{\lambda}(T, \lambda) d\lambda = \frac{8\pi\kappa_{0}\lambda_{0}^{\beta}(kT)^{4+\beta}}{c^{2+\beta}h^{3+\beta}} \Gamma(4+\beta)\zeta(4+\beta), \qquad (\text{I.75}) $$

where $\Gamma$ is the *gamma function*, and $\zeta$ is the *Riemann zeta function*.

**Wien's law** states that the emission peak of $B_v(T, \lambda)$ is located at $\lambda_{\text{max}}(T) = 5.0996 \times 10^3/\text{T} \, \mu\text{m}$. For a MBB, the wavelength peak of $\nu\epsilon_v(\beta, T, \lambda)^9$ is located at:

$$ \lambda_{\text{max}}(\beta, T) = \frac{hc}{kT(4+\beta) + W(-(4+\beta)\exp[-(4+\beta)])}, \qquad (\text{I.76}) $$

where W is the *Lambert W function*.

<figure><img src="image_128.png" /></figure>

<footnote n="9">9. In general, we prefer displaying $\nu f_v$ quantities than simply $f_v$ or $f_\lambda$, as it represents better the energy balance.</footnote>
<figure><img src="image_129.png" /></figure>

### I.2.4.2 Equilibrium Heating

Eq. (I.71) expresses the general emissivity of a grain at thermal equilibrium with the radiation field. To use this formula, we need to determine the equilibrium or steady-state temperature of the grain. This is simply performed by equating the absorbed and emitted powers. It is convenient to define the mean intensity of the ISRF:

$$J_v(\lambda) = \frac{1}{4\pi} \iint_{\text{sphere}} I_v(\lambda, \Omega) \, d\Omega, \qquad (I.77)$$

which is simply the specific intensity from the stars, averaged over solid angle. The power absorbed by the grain is:

$$P_{\text{abs}}(a) = \iint_{\text{sphere}} \int_{0}^{\infty} J_{v}(v) \times \pi a^{2} Q_{\text{abs}}(a, v) \, dv \, d\Omega = \int_{0}^{\infty} 4\pi J_{v}(v) \times \pi a^{2} Q_{\text{abs}}(a, v) \, dv. \qquad (I.78)$$

Similarly, the emitted power is:

$$P_{\text{em}}(a, T) = \iint_{\text{sphere}} \int_{0}^{\infty} B_{v}(T, v) \times \pi a^{2} Q_{\text{abs}}(a, v) \, dv \, d\Omega = \int_{0}^{\infty} 4\pi B_{v}(T, v) \times \pi a^{2} Q_{\text{abs}}(a, v) \, dv. \qquad (I.79)$$

The equilibrium temperature, T<sub>eq</sub>, is then simply the numerical solution to P<sub>abs</sub>(a) = P<sub>em</sub>(a, T<sub>eq</sub>). Several quantities can be precomputed to simplify this task.

**Planck average.** The Planck average of a grain is defined as:

$$\langle Q \rangle_{\text{P}}(a, T) = \frac{\pi}{\sigma T^4} \int_{0}^{\infty} Q_{\text{abs}}(a, v) B_{v}(T, v) \, dv. \qquad (I.80)$$

This quantity needs to be computed only once for a range of temperatures. Eq. (I.79) then simplifies: P<sub>em</sub>(a, T) = 4πa<sup>2</sup>σT<sup>4</sup>⟨Q⟩<sub>P</sub>(a, T). Since interstellar grains emit predominantly in the IR, where the approximation of Eq. (I.73) is valid, we can derive an analytical expression of Eq. (I.80), knowing that σ ≡ 2π<sup>5</sup>k<sup>4</sup>/15h<sup>3</sup>c<sup>2</sup>:

$$\langle Q \rangle_{\text{P}}(a, T) = \frac{15}{\pi^4} Q_{\text{abs}}(a, \lambda_0) \Gamma(4 + \beta) \zeta(4 + \beta) \left( \frac{\lambda_0 k T}{hc} \right)^{\beta}. \qquad (I.81)$$

We show the Planck average of typical grains in Fig. I.27. We can see that ⟨Q⟩<sub>P</sub> is almost independent of radius for grains smaller than 0.1 µm.

**ISRF-averaged efficiency.** The ISRF-averaged efficiency is the equivalent of the Planck average for the absorbed power:

$$\langle Q \rangle_{\star}(a) = \frac{\pi}{J_{\star}} \int_{0}^{\infty} Q_{\text{abs}}(a, v) J_{v}(v) \, dv, \qquad (I.82)$$

where J<sub>★</sub> = π ∫<sub>0</sub><sup>∞</sup> J<sub>v</sub>(v) dv. This quantity is less general than Eq. (I.81) as it needs to be evaluated for each particular shape of the ISRF. Eq. (I.78) then simplifies: P<sub>abs</sub>(a) = 4πa<sup>2</sup>J<sub>★</sub>⟨Q⟩<sub>★</sub>(a). The equilibrium temperature is thus the solution to: σT<sup>4</sup>⟨Q⟩<sub>P</sub>(a, T) = J<sub>★</sub>⟨Q⟩<sub>★</sub>(a).

**The diffuse ISRF.** The diffuse ISRF of the MW has been modeled by Mathis et al. (1983). It is represented in Fig. I.28. This ISRF is commonly used to describe grain heating (i.e. how much power a grain absorbs) in the MW, and also in nearby galaxies. Most of the heating is provided by the stellar component, as the integrand in Eq. (I.78) is J<sub>v</sub>Q<sub>abs</sub> ∝ J<sub>v</sub>/λ<sup>2</sup>. The long wavelengths have a negligible weight. This stellar ISRF can be scaled by a dimensionless factor U, to account for variations of the stellar density. This scaling factor is not totally realistic, as regions with high radiation densities (U ≳ 10<sup>3</sup>) usually are star-forming regions, containing young star associations. The UV bump of the stellar ISRF, corresponding to these young stars in Fig. I.28, would be dominating the emission, while the NIR bump, corresponding to older stellar populations, would be, at first order constant. This is however not very important for grains at thermal equilibrium, as their spectrum depends only on the total absorbed power, given by ⟨Q⟩<sub>★</sub>.

<figure><img src="image_130.png" /></figure>
<figure><img src="image_131.png" /></figure>

<figure><img src="image_132.png" /><figcaption>FIGURE I.27 – Planck averages of graphite and silicates. We show the result of Eq. (I.80) applied to the graphite (a) and silicates (b) of Draine (2003b,c), for several radii, *a*. We overplot the approximation of Eq. (I.81) in grey. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_133.png" /><figcaption>FIGURE I.28 – *Diffuse Galactic ISRF.* This black line represents the average ISRF experienced in the diffuse ISM of the MW. The stellar component (yellow) has been modeled by Mathis et al. (1983). The UV and NIR bumps are the contributions by young and old stars, respectively. This ISRF represents the neutral ISM, there is therefore no emission shortward the Lyman break ($\lambda_{LyC}$ = 0.0912 µm). This stellar ISRF can be scaled by the factor U. We have represented the U = 10 case in grey. The diffuse dust emission is shown in red. The original Mathis et al. (1983) work did not have the constraints we have today on this component. The emission represented here is the Jones et al. (2017) model, for a typical hydrogen column density $N_H = 10^{21} \text{ cm}^{-2}$. The blue component is the Cosmic Microwave Background (CMB), which is a perfect black body at $T_{CMB}(z=0) = 2.73 \text{ K}$ (Mather et al., 1994). At higher redshift, $z$, the temperature of this component is $T_{CMB}(z) = (1+z) \times 2.73 \text{ K}$. The Cosmic Infrared Background (CIB; e.g. Dole et al., 2006) is not represented here as its Spectral Energy Distribution (SED) is similar to the dust emission, and is slightly lower. Licensed under CC BY-SA 4.0.</figcaption></figure>
**Equilibrium temperatures.** The equilibrium temperature of typical grains is shown in Fig. I.29, as a function of U. Assuming the heating is solely provided by the stellar component in Fig. I.28, $J_*(U) = U \times J_*(1)$ and $\langle Q \rangle_*(U, a) = \langle Q \rangle_*(1, a)$. In this case, the integrated mean intensity is:

$$ \int_{0}^{\infty} 4\pi J_{v}(U = 1, v) dv = 4 \times J_{*}(1) = 2.2 \times 10^{-5} \text{ W/m}^{2}. \qquad (I.83) $$

The equilibrium temperature is thus:

$$ T_{\text{eq}}(U, a) = \underbrace{\left( \frac{\pi^4 J_*(1) \langle Q \rangle_*(a)}{\sigma Q_{\text{abs}}(a, \lambda_0) \Gamma(4+\beta) \zeta(4+\beta)} \right)^{1/(4+\beta)}}_{\text{weakly dependent on } a} \left( \frac{hc}{k\lambda_0} \right)^{\beta/(4+\beta)} U^{1/(4+\beta)}. \qquad (I.84) $$

For grains smaller than $a \approx 0.1$ µm, we thus have $T_{\text{eq}}(U) \propto U^{1/(4+\beta)}$. We show the equilibrium temperature of graphite and silicates, varying U in Fig. I.29. We see that:

* for graphite, $T_{\text{eq}}^{\text{gra}}(U) \approx U^{1/6} \times 20$ K;

* for silicates, $T_{\text{eq}}^{\text{sil}}(U) \approx U^{1/6} \times 17.5$ K.

Interstellar grains of a given homogeneous composition, at thermal equilibrium with the radiation field, mostly have the same temperature.

<figure><img src="image_134.png" /><figcaption>FIGURE I.29 – Grain equilibrium temperatures. We show the equilibrium temperatures derived by equating Eq. (I.78) and Eq. (I.79) for the graphite of Laor & Draine (1993) and the silicates of Draine (2003b,c). For grains smaller than ≈ 20 nm, this temperature does not have reality, as these grains are out of equilibrium with the ISRF. The grains are heated by the stellar ISRF of Mathis et al. (1983), scaled by the factor U. We see that grains smaller than ≈ 0.1 µm roughly have the same temperature. This is because these grains are essentially in the Rayleigh regime over most of the visible-to-NIR range. In this regime, the dashed grey line is a good approximation. Licensed under CC BY-SA 4.0.</figcaption></figure>

### I.2.4.3 Stochastic Heating

**Absorption and cooling times.** Not all grains are at thermal equilibrium with the ISRF. To determine if this is the case, we need to estimate the photon absorption rate of the grain:

$$ \frac{1}{\tau_{\text{abs}}(U, a)} = \int_{0}^{\infty} \pi a^{2} Q_{\text{abs}}(a, v) \frac{4\pi J_{v}(v)}{h\nu} dv \propto a^{3}U, \qquad (I.85) $$
<figure><img src="image_135.png" /></figure>

where the proportionality has been derived using Eq. (I.46). The grain absorption timescale, $\tau_{abs}$, gives the average time between two photon absorptions. We also need to estimate the cooling rate of the grain:

$$ \frac{1}{\tau_{\text{cool}}(a, T)} = \frac{P_{\text{em}}(a, T)}{H(T)} \propto T^{3-n+\beta}, \qquad (\text{I.86}) $$

where $H(T)$ is the enthalpy of the grain at temperature T:

$$ H(T) = \int_{0}^{T} C(T') dT'. \qquad (\text{I.87}) $$

It is the vibrational energy content of the grain. The proportionality of Eq. (I.86) has been derived from Eq. (I.81) for $P_{em}$, and from the low-temperature behaviour of $H(a, T) \propto a^3 T^{n+1}$ (Eq. I.67; $n=3$ corresponding to the three-dimensional Debye model). The cooling time, $\tau_{cool}$, is independent of the grain size.

**The temperature fluctuations.** If $\tau_{abs} \gtrsim \tau_{cool}$, the grain has the time to significantly cool down between two photon absorptions. Its temperature is thus changing with time. This is represented on the simulation in Fig. I.30.a, as follows.

1. A grain, with $a = 2$ nm, starts at $T = T_{CMB}$.

2. It then receives a photon after $\approx 6$ h, which causes its temperature to spike to $T_s \approx 100$ K. The value of $T_s$ is such that $H(T_s) - H(T_{CMB}) = h\nu_s$, where $h\nu_s$ is the energy of the incident photon. It is the only photon it receives within the 50 h displayed here. Its absorption time is indeed $\tau_{abs} \approx 140$ h.

3. The grain then cools down by radiating. The heat capacity we have used here (Draine & Li, 2001) has $n \approx 2$. The cooling time is thus $\tau_{cool} \propto T^{-3}$. The cooling time is short when the grain is hot, but decreases with the temperature. Such a grain spends most of its time at low temperatures.

In practice, we do not see the emission of the grain varying with time, as observations encompass a statistical number of grains, with different time histories. What we observe is an ergodic<fnref n="10" /> average: small grains appear to have a temperature distribution, represented in Fig. I.30.b for $a = 2$ nm. When the radius of the grain increases, the absorption time decreases as $\tau_{abs} \propto a^{-3}$, as shown in the remaining panels of Fig. I.30. The number of temperature spikes increases, as the grain being larger, it intercepts more photons. The temperature of the spikes also decreases with $a$, as the grain stores the energy of a single photon in a larger number of phonon modes. For large enough grains (Fig. I.30.g), the temperature fluctuations become negligible. The grain appears to have a single temperature; its temperature distribution tends toward a Dirac distribution (Fig. I.30.h): it has reached thermal equilibrium. Fig. I.31 shows the same type of simulations, fixing the radius of the grain, and varying the starlight intensity. In this case, the heating rate increases linearly with U (downward in Fig. I.31).

**Out-of-equilibrium emission.** At each time, in the simulations of Figs. I.30–I.31, the emission of the grain is: $\epsilon_v(a, v, t) = 3\pi/\rho \times Q_{abs}(a, v)/a \times B_v(T(t), v)$. To average over time, we simply need to integrate over the temperature distribution:

$$ \epsilon_v(a, v) = \frac{3\pi Q_{\text{abs}}(a, v)}{\rho a} \int_{0}^{\infty} B_v(T, v) \frac{dP(T, a)}{dT} dT. \qquad (\text{I.88}) $$

The left panels of Fig. I.32 show the temperature distributions of PAHs, graphite and silicates of different sizes. The corresponding emission spectra, computed using Eq. (I.88), are shown in the right panels. We see that the smallest grains fluctuate to the highest temperatures. Their emission is thus the broadest, and extends to the shortest wavelengths. An important difference between equilibrium

<figure><img src="image_136.png" /></figure>

<footnote n="10">10. The ergodicity is a principle stating that the steady-state statistical distribution of the properties of a large number of identical particles is the average of their properties over time.</footnote>
<figure><img src="image_137.png" /></figure>

<figure><img src="image_138.png" /><figcaption>FIGURE 1.30 – *Temperature fluctuations of grains with different radii.* The left panels show the time variation of the temperature of silicate grains (Draine, 2003b,c), exposed to the Mathis et al. (1983) ISRF with $U = 1$. The radius of the grain, $a$, increases downward. The right panels show the corresponding probability distribution of the temperature. These simulations were performed using the Draine & Anderson (1985) Monte-Carlo method. See Draine (2003a) for a similar simulation with graphite. Licensed under CC BY-SA 4.0.</figcaption></figure>

<figure><img src="image_139.png" /></figure>
<figure><img src="image_140.png" /></figure>

<figure><img src="image_141.png" /><figcaption>FIGURE I.31 – Temperature fluctuations of grains with different starlight intensities. This is a variation on Fig. I.30. The grain radius is constant (a = 3 nm), and the starlight intensity, U, increases downward. Licensed under CC BY-SA 4.0.</figcaption></figure>
and stochastic heating is that the emission spectrum of small grains depends not only on the intensity of the ISRF, but also on its hardness. The latter can be quantified with the mean energy of stellar photons:

$$ \langle hv \rangle \equiv \frac{\int_{0}^{\infty} J_{v}(v) dv}{\int_{0}^{\infty} \frac{J_{v}(v)}{hv} dv}. \qquad (I.89) $$

This parameter determines the average temperature spikes. This is the reason why stochastic heating is sometimes referred to as single photon heating, transient heating or quantum heating. The ISRF intensity roughly scales with the emissivity, but does not affect its spectral shape.

**Numerical methods.** To compute the emission spectrum of an out-of-equilibrium grain, we need to calculate its temperature distribution, dP/dT (Eq. I.88). This is done numerically. Several methods can be found in the literature.

**The Monte-Carlo method** (Draine & Anderson, 1985) consists in simulating the temporal evolution of the grain, by drawing random stellar photons. This is the method we have used in Figs. I.30 – I.31. This method is easy to implement, but it is not the most numerically efficient.

**Solving the integral equation** governing dP/dT (Desert et al., 1986). This method is efficient and is used by the DustEM code (Compiègne et al., 2011).

**The transition matrix method** (Guhathakurta & Draine, 1989) consists in building a squared matrix whose elements are the probability per unit time that a grain will transit from a state to another. The row and columns of this matrix correspond to the final and initial energy bins. This matrix can then be diagonalized to compute dP/dT. This is the method we have implemented in our code. It has been used in Fig. I.32. We have implemented an adaptive grid in energy, represented by the dots in the left panels of Fig. I.32. The grid is refined to ensure the accuracy of the emission spectrum. We can see that more points are needed at high temperatures.

**ISRF moment approximations** (Natale et al., 2015) consists in interpolating a precomputed grid, characterizing the ISRF by its first two moments: its intensity, U, and the mean energy of the photons, $\langle hv \rangle$. This method is not exact, but it is fast enough to be implemented in radiative transfer simulations.

**Equilibrium criterion.** A simple criterion to determine if a grain is at thermal equilibrium with the ISRF consists in comparing its equilibrium enthalpy with the average energy of a stellar photon. Fig. I.33 compares these two quantities for graphite and silicates, varying a and U.

A grain is at thermal equilibrium if $H(a, T_{eq}) \gg \langle hv \rangle$.

**The transition radius** between stochastically heated and equilibrium grains can therefore be estimated as the radius for which $H(a_t, T_{eq}) \approx 20 \times \langle hv \rangle$. This value is indicative as the transition to equilibrium is a smooth, continuous process. These transition radii are the vertical dashed lines in Fig. I.33. The mean stellar photon energy of the Mathis et al. (1983) ISRF is a constant: $\langle hv \rangle \approx 1.05$ eV. The equilibrium enthalpy behaves as $H(a_t, T_{eq}) \propto a_t^3 T_{eq}^{n+1}$. Since $T_{eq} \propto U^{1/(4+\beta)}$, we have $a_t \propto U^{-(n+1)/(3(4+\beta))} \approx U^{-1/6}$ ($n=2$ and $\beta=2$ for the grains in Fig. I.33). The following approximations provide good fits:

* for graphite, $a_t^{\text{gra}}(U) \approx 22 \text{ nm} \times U^{-1/6}$;

* for silicates, $a_t^{\text{sil}}(U) \approx 15 \text{ nm} \times U^{-1/6}$.

**Role of the CMB:** note that, contrary to a common misconception, the smallest grains are not in thermal equilibrium with the CMB. For instance, a 3 Å silicate has an enthalpy at 2.7 K of $H(T_{\text{CMB}}) \approx 0.5$ µeV, while the average photon energy received from the CMB is $\langle hv_{\text{CMB}} \rangle \approx 1$ meV.
<figure><img src="image_142.png" /></figure>

<figure><img src="image_143.png" /><figcaption>FIGURE 1.32 – Stochastically heated grains. The left panels show the temperature distribution of grains with different radii. The dots represent the sampling of our adaptative energy grid. The PAHs are a mixture of neutral (50%) and ionized (50%) molecules (Draine & Li, 2007), the graphite and silicates are from Draine (2003b,c). The right panels show the corresponding emissivities. Licensed under CC BY-SA 4.0.</figcaption></figure>
<figure><img src="image_144.png" /></figure>

The effect of considering photons with $\lambda > 1000$ µm as a source of continuous heating (following Guhathakurta & Draine, 1989), creates an artificial minimum temperature close to the actual CMB temperature. This is the bump peaking at $\lambda \approx 1$ mm for the smallest grains in panels (e) and (f) of Fig. I.32. This is an artefact due to this approximation. Fortunately, this artefact does not affect the emitted spectrum, integrated over the size distribution, in any detectable way.

<figure><img src="image_145.png" /><figcaption>FIGURE I.33 – Transition radius for stochastically heated grains. These two panels show the enthalpy at the hypothetical equilibrium temperature, for the graphite of Laor & Draine (1993) and the silicates of Draine (2003b,c). The horizontal dashed grey line represents the $20 \times \langle hv \rangle$ level, for the Mathis et al. (1983) ISRF. The enthalpies above this line correspond to $H(T_{eq}) \gg \langle hv \rangle$, that is to equilibrium grains. The vertical dashed color lines, show the transition radius for each U. Left of this line, grains are stochastically heated. Licensed under CC BY-SA 4.0.</figcaption></figure>

### I.2.4.4 Collisional Heating

In addition to photon absorption, collisions with gas particles can, in specific conditions, contribute to dust heating. Obviously, this will happen when the temperature of the gas is the hottest, such as in a plasma. The collision rate of gas particles, following a Maxwell-Boltzmann distribution, with a dust grain is:

$$ \frac{1}{\tau_{\text{coll}}} \approx \underbrace{\sqrt{\frac{8kT}{\pi m}}}_{\text{mean velocity}} \underbrace{\frac{1}{n\pi a^2}}_{\text{mean free path}}, \qquad (\text{I.90}) $$

where $n$ is the density of the gas, and $m$, the mass of the particles. Assuming that the protons and the electrons are thermalized, the ratio of their collision rates is $\tau_{\text{coll}}(e^-)/\tau_{\text{coll}}(H^+) = \sqrt{m_e/m_p} \approx 0.02$ (cf. Table B.2). The collisions with the protons can thus be neglected.

**Electronic heating rate.** The Maxwell-Boltzmann distribution of the electrons of energy, $E$, can be written:

$$ f(E, T) = \frac{2}{\sqrt{\pi}} \sqrt{\frac{E}{(kT)^3}} \exp\left(-\frac{E}{kT}\right). \qquad (\text{I.91}) $$

<figure><img src="image_146.png" /></figure>
<figure><img src="image_147.png" /></figure>

This distribution is normalized: $\int_{0}^{\infty} f(E, T) dE = 1$. It is displayed in Fig. I.34 and compared to the stellar radiation field. The collisional cross-section is very poorly constrained. At low energies, it should be close to the geometric cross-section, $\pi a^2$. However, at high energies, electrons can pass thought the grain. Dwek (1986) proposed the following cross-section:

$$ \sigma_{\text{coll}}(a, E) = \pi a^2 \times \begin{cases} 1 & \text{for } E < E_{\star}(a) \\ 1 - \left[ 1 - \left( \frac{E_{\star}(a)}{E} \right)^{3/2} \right]^{2/3} & \text{for } E \ge E_{\star}(a), \end{cases} \qquad (\text{I.92}) $$

where $E_{\star}(a)$ is the threshold energy. According to the fit of experimental data shown in Fig. 1 of Dwek (1987), this threshold energy is (cf. the discussion in Bocchio et al., 2013):

$$ \left\{ \begin{array}{ll} E_{\star}^{\text{car}}(a) & \approx 10 \, \text{keV} \times \left( \frac{a}{1 \, \mu\text{m}} \right)^{2/3} & \text{for carbonaceous} \\ E_{\star}^{\text{sil}}(a) & \approx 14 \, \text{keV} \times \left( \frac{a}{1 \, \mu\text{m}} \right)^{2/3} & \text{for silicates.} \end{array} \right. \qquad (\text{I.93}) $$

Assuming the grains are at rest, the collision rate for a given energy, E, is now:

$$ \frac{1}{\tau_{\text{coll}}(a, E)} = n \times \sigma_{\text{coll}}(a, E) \times \nu(E), \qquad (\text{I.94}) $$

where the velocity of an electron with energy E is:

$$ \nu(E) = \sqrt{\frac{2E}{m}}. \qquad (\text{I.95}) $$

The collisional power received by the grain is finally the integral of the energy deposit per unit time, $E/\tau_{\text{coll}}$, over the whole Maxwell-Boltzmann distribution:

$$ P_{\text{coll}}(a, n, T) = \int_{0}^{\infty} \frac{E}{\tau_{\text{coll}}(a, E)} f(E, T) dE. \qquad (\text{I.96}) $$

**Cases where collisions dominate the heating.** Coronal plasmas, that can be found in the *Hot Ionized Medium* (HIM; cf. Table III.6) of the MW or the halo of elliptical galaxies, have typical temperatures of $T \approx 10^6 - 10^7$ K. This gas has been heated by successive *SuperNova* (SN) blast waves. It has a low density ($n \approx 10^{-3} - 10^{-2}$ cm$^{-3}$), but can have a large filling factor (it fills 50% of the volume of the MW). At these temperatures, it is responsible for a diffuse X-ray emission. When dust grains are present in such a gas, collisions can dominate the heating, depending on the balance between the photon and electron energy densities (cf. Fig. I.34).

**The collisional heating rate** is displayed as a function of the grain radius in Fig. I.35. This figure basically shows that, if a grain is in a coronal plasma, collisional heating dominates for $U < 0.01$, and photon heating dominates if $U > 10$. In between, both can play a role, depending on n and T.

**The transition radius** between stochastically-heated and equilibrium grains can be computed similarly to Eq. (I.33), using the heating rate of Fig. I.35. This time, the single heating events are due to electron collisions, with average energies: $\langle E \rangle = 3/2kT \approx 0.13 \text{ keV} \times T/10^6$ K. They are much higher than the typical ISRF photon energy, $\langle h\nu \rangle \approx 1$ eV. The stochastic heating will be the most efficient for the lowest densities and highest temperatures. For the extreme case, $n = 10^{-3}$ cm$^{-3}$ and $T = 10^7$ K, the transition radii are: $a_t^{\text{gra}} \approx 58$ nm and $a_t^{\text{sil}} \approx 35$ nm.

<figure><img src="image_148.png" /></figure>
<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

FIGURE 1.34 – Photon and electron energy densities. The blue and cyan curves show the energy per unit volume ($u_v = 4\pi cJ_v$) of the stellar photons, for U = 1 and U = 10. The electron energy density is the Maxwell-Boltzmann distribution (Eq. I.91): $n \times f(E)$. We have plotted it for densities and temperatures typical of coronal plasmas: $n = 10^{-3}–10^{-2} \text{ cm}^{-3}$ and T = $10^6–10^7$ K. Licensed under CC BY-SA 4.0.

<figure><img src="image_151.png" /></figure>

FIGURE 1.35 – Collisional heating rate. In each panel, the cyan, blue, purple and magenta lines show the collisional heating rates from Eq. (I.96), for different values of *n* and T. We compare the stellar photon heating rate in yellow (U = 0.01) and orange (U = 10). The area colored in yellow corresponds to U < 0.01, where photon heating is negligible. The area colored in orange corresponds to U > 10, where photon heating dominates. Licensed under CC BY-SA 4.0.
<figure><img src="image_152.png" /></figure>

# Chapter II

# Dust Observables and Models

*All models are wrong, but some are useful.*

(George E. P. Box; Box, 1979)

## Contents

<table><tr><td>II.1</td><td>A Brief History of Interstellar Dust Studies</td><td>53</td></tr><tr><td>II.1.1</td><td>The Challenges of Observing Interstellar Regions</td><td>55</td></tr><tr><td>II.1.1.1</td><td>Limitations Due to the Atmosphere</td><td>55</td></tr><tr><td>II.1.1.2</td><td>Historical Ground-Based Observatories</td><td>55</td></tr><tr><td></td><td>The meter-class visible telescopes.</td><td>55</td></tr><tr><td></td><td>The first MIR telescopes.</td><td>55</td></tr><tr><td></td><td>The submillimeter observatories.</td><td>55</td></tr><tr><td>II.1.1.3</td><td>Airborne Observatories</td><td>56</td></tr><tr><td></td><td>Balloons.</td><td>56</td></tr><tr><td></td><td>Airplanes.</td><td>57</td></tr><tr><td>II.1.1.4</td><td>Space Telescopes</td><td>57</td></tr><tr><td></td><td>IRAS.</td><td>57</td></tr><tr><td></td><td>COBE.</td><td>57</td></tr><tr><td></td><td>ISO.</td><td>58</td></tr><tr><td></td><td>Spitzer.</td><td>58</td></tr><tr><td></td><td>AKARI.</td><td>58</td></tr><tr><td></td><td>WISE.</td><td>59</td></tr><tr><td></td><td>Herschel.</td><td>59</td></tr><tr><td></td><td>Planck.</td><td>59</td></tr><tr><td></td><td>The JWST.</td><td>59</td></tr><tr><td></td><td>UV and X-ray Satellites.</td><td>61</td></tr><tr><td>II.1.1.5</td><td>Grain-Collecting Spacecrafts</td><td>61</td></tr><tr><td>II.1.2</td><td>Chronology of the Main Breakthroughs</td><td>62</td></tr><tr><td>II.1.2.1</td><td>Obscuration and Dimming of Starlight</td><td>62</td></tr><tr><td></td><td>The discovery of dark nebulae.</td><td>62</td></tr><tr><td></td><td>The reddening of starlight.</td><td>62</td></tr><tr><td>II.1.2.2</td><td>The Dust Continuum</td><td>63</td></tr><tr><td></td><td>The Shape of the extinction curve.</td><td>63</td></tr><tr><td></td><td>Polarization by dichroic extinction.</td><td>64</td></tr><tr><td></td><td>Dust emission.</td><td>64</td></tr><tr><td>II.1.2.3</td><td>Identification of Dust Features</td><td>64</td></tr><tr><td></td><td>Silicates.</td><td>65</td></tr></table>

<figure><img src="image_153.png" /></figure>
<figure><img src="image_154.png" /></figure>

<table>
  <tbody>
    <tr>
      <td>Carbonaceous grains.</td>
      <td>65</td>
    </tr>
    <tr>
      <td>Ices.</td>
      <td>65</td>
    </tr>
    <tr>
      <td>X-ray edges.</td>
      <td>65</td>
    </tr>
    <tr>
      <td>II.1.2.4 Dusty Epiphenomena</td>
      <td>66</td>
    </tr>
    <tr>
      <td>Diffuse Interstellar Bands.</td>
      <td>66</td>
    </tr>
    <tr>
      <td>Extended Red Emission.</td>
      <td>66</td>
    </tr>
    <tr>
      <td>Spinning Grains.</td>
      <td>66</td>
    </tr>
    <tr>
      <td>II.1.2.5 Dust Models</td>
      <td>66</td>
    </tr>
    <tr>
      <td>First models.</td>
      <td>66</td>
    </tr>
    <tr>
      <td>Calculation of the optical properties.</td>
      <td>67</td>
    </tr>
    <tr>
      <td>Elemental depletions.</td>
      <td>67</td>
    </tr>
    <tr>
      <td>Modern panchromatic models.</td>
      <td>67</td>
    </tr>
    <tr>
      <td>II.2 The Current Empirical Constraints</td>
      <td>68</td>
    </tr>
    <tr>
      <td>II.2.1 Extinction</td>
      <td>68</td>
    </tr>
    <tr>
      <td>II.2.1.1 UV-to-NIR Extinction</td>
      <td>70</td>
    </tr>
    <tr>
      <td>The extinction magnitude.</td>
      <td>70</td>
    </tr>
    <tr>
      <td>The selective extinction.</td>
      <td>71</td>
    </tr>
    <tr>
      <td>Measuring extinction.</td>
      <td>71</td>
    </tr>
    <tr>
      <td>UV-visible scattering.</td>
      <td>71</td>
    </tr>
    <tr>
      <td>II.2.1.2 MIR Extinction</td>
      <td>72</td>
    </tr>
    <tr>
      <td>The MIR continuum.</td>
      <td>72</td>
    </tr>
    <tr>
      <td>Silicate features.</td>
      <td>73</td>
    </tr>
    <tr>
      <td>Ices.</td>
      <td>73</td>
    </tr>
    <tr>
      <td>II.2.1.3 X-Rays</td>
      <td>73</td>
    </tr>
    <tr>
      <td>X-ray halos.</td>
      <td>73</td>
    </tr>
    <tr>
      <td>X-ray absorption edges.</td>
      <td>74</td>
    </tr>
    <tr>
      <td>II.2.1.4 Dichroic Extinction</td>
      <td>75</td>
    </tr>
    <tr>
      <td>II.2.1.5 Diffuse Interstellar Bands</td>
      <td>75</td>
    </tr>
    <tr>
      <td>II.2.2 Emission</td>
      <td>76</td>
    </tr>
    <tr>
      <td>II.2.2.1 Infrared Continuum and Features</td>
      <td>76</td>
    </tr>
    <tr>
      <td>Observations of the diffuse ISM.</td>
      <td>76</td>
    </tr>
    <tr>
      <td>MIR features.</td>
      <td>76</td>
    </tr>
    <tr>
      <td>II.2.2.2 Polarized Emission</td>
      <td>77</td>
    </tr>
    <tr>
      <td>II.2.2.3 Non-Thermal Emission</td>
      <td>77</td>
    </tr>
    <tr>
      <td>Spinning Grains.</td>
      <td>77</td>
    </tr>
    <tr>
      <td>Photoluminescence.</td>
      <td>77</td>
    </tr>
    <tr>
      <td>II.2.3 Elemental Abundances in Grains</td>
      <td>78</td>
    </tr>
    <tr>
      <td>II.2.3.1 Measuring ISM Abundances</td>
      <td>78</td>
    </tr>
    <tr>
      <td>Solar abundances.</td>
      <td>78</td>
    </tr>
    <tr>
      <td>Present-day Solar abundances.</td>
      <td>78</td>
    </tr>
    <tr>
      <td>II.2.3.2 Depletions</td>
      <td>79</td>
    </tr>
    <tr>
      <td>The depletion strength.</td>
      <td>79</td>
    </tr>
    <tr>
      <td>Volatile and refractory elements.</td>
      <td>80</td>
    </tr>
    <tr>
      <td>Inferred dust composition.</td>
      <td>80</td>
    </tr>
    <tr>
      <td>II.2.4 Direct Measures</td>
      <td>81</td>
    </tr>
    <tr>
      <td>II.2.4.1 Meteorite Inclusions</td>
      <td>82</td>
    </tr>
    <tr>
      <td>Grain identification.</td>
      <td>82</td>
    </tr>
    <tr>
      <td>Limitations.</td>
      <td>82</td>
    </tr>
    <tr>
      <td>II.2.4.2 Interplanetary Dust</td>
      <td>83</td>
    </tr>
    <tr>
      <td>ISD flux and cometary dust.</td>
      <td>83</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_155.png" /></figure>
<table><tbody><tr><td>Micrometeorites.</td><td>83</td></tr><tr><td>II.2.4.3 Laboratory Measurements</td><td>83</td></tr><tr><td>Spectroscopic characterization.</td><td>83</td></tr><tr><td>Grain reactivity and evolution.</td><td>84</td></tr><tr><td>II.3 State-of-the-Art Dust Models</td><td>85</td></tr><tr><td>II.3.1 Composition and Size Distributions of Different Models</td><td>85</td></tr><tr><td>II.3.1.1 Diversity in Composition</td><td>85</td></tr><tr><td>Inherent degeneracies of dust models.</td><td>85</td></tr><tr><td>Common compositional choices.</td><td>85</td></tr><tr><td>II.3.1.2 Difference in Size Distributions</td><td>86</td></tr><tr><td>Origin of the Size Distribution.</td><td>86</td></tr><tr><td>Comparison Between Different Models.</td><td>87</td></tr><tr><td>II.3.2 The Model Properties</td><td>87</td></tr><tr><td>II.3.2.1 Extinction and Emission</td><td>88</td></tr><tr><td>The opacity.</td><td>88</td></tr><tr><td>The SED.</td><td>88</td></tr><tr><td>II.3.2.2 The Fitted Constraints</td><td>89</td></tr><tr><td>The THEMIS model.</td><td>89</td></tr><tr><td>Discussion of the fit.</td><td>89</td></tr><tr><td>II.3.3 Some Useful Quantities</td><td>89</td></tr><tr><td>II.3.3.1 Grain Sizes, Areas and Masses</td><td>89</td></tr><tr><td>For the MRN size distribution.</td><td>89</td></tr><tr><td>The case of the THEMIS model.</td><td>92</td></tr><tr><td>Mass fraction of small grains.</td><td>92</td></tr><tr><td>Dustiness and other ratios.</td><td>92</td></tr><tr><td>II.3.3.2 Opacity and Emissivity</td><td>93</td></tr><tr><td>Optical properties.</td><td>93</td></tr><tr><td>Heating regimes.</td><td>93</td></tr><tr><td>Emissivity.</td><td>93</td></tr></tbody></table>

The present chapter is intended to provide a comprehensive overview of the current state of the field. It discusses both the observables that can be used to constrain dust properties, and the current state-of-the-art models. It bridges the basic physical knowledge of Chap. I with their current application.

## II.1 A Brief History of Interstellar Dust Studies

The History of our progressive understanding of ISD is driven by the successive technological innovations that allowed us to shed light on its nature.

**Telescopes** allowed *deep-sky* observations already through the XVIIIth century (e.g. Wilson, 2007). Charles MESSIER’s catalog of “nebulae” was first established in 1774. The quality and size of mirrors increased through the XIXth century, and at the beginning of the XXth century, the first meter-class telescopes, with silvered-glass mirrors<fnref n="1" />, were built. The Mount Wilson Observatory was founded in 1904, and its 1.5-m Hale telescope was commissioned in 1908.

**Photographic plates** turned astronomical observations into a reproducible empirical science. They required long exposures and reliable tracking of the sky’s apparent rotation. Photography was invented by Nicéphore NIÉPCE in the 1820’s. His associate, Louis DAGUERRE, perfected the

<footnote n="1">1. Before that, telescope mirrors were made of the lower optical quality speculum metal alloy. The process of silvering glass mirrors was invented by Léon FOUCAULT in the 1860’s.</footnote>
<figure><img src="image_156.png" /><figcaption>(a) First photograph of Orion (1880)</figcaption></figure>

<figure><img src="image_157.png" /><figcaption>(b) Sir William Herschel</figcaption></figure>

<figure><img src="image_158.png" /><figcaption>(c) Reenactment</figcaption></figure>

**FIGURE II.1 – Early ISM and IR astronomy.** Panel (a) shows the 1880 picture of the Orion nebula taken at the Lick observatory, by Henry DRAPER. Panel (b) represents Sir William HERSCHEL measuring the SED of the Sun. We can see in the background his 1.2 m speculum mirror telescope. Panel (c) shows a simple recreation of his experiment with commercial thermometers. We see that the temperature is the highest on the last thermometer, after the red. Credit: (a) Henry DRAPER, public domain; (b) El Sofista, not licensed; (c) courtesy of NASA/JPL-Caltech.

technique and commercialized the first daguerreotype<fnref n="2" /> in 1839. The first attempts to capture images of the sky (the Moon and the Sun), were performed using this invention, in the 1840's. Emulsion plates were substituted to daguerreotypes, and the first deep-sky picture (the Orion nebula; Fig. II.1.a) was taken by Henry DRAPER in 1880 (Barker, 1888).

**Solid-state physics** was given a strong impulse, after World War II, by the prospective development of electronics (Martin, 2013). The transistor was invented in 1947, at Bell laboratories in New Jersey. This impulse led to the development of technical and conceptual tools for the optical and electronic properties of solids that our field would benefit from.

**The introduction of computers** revolutionized all scientific fields. The principle of the computer was laid out in Alan TURING's seminal 1937 article (Turing, 1937). The first computers were developed during World War II to break the German encryption codes (e.g. McGrayne, 2011). They started to be used in astrophysics during the 1950's, to compute stellar structures. Before that, some calculations were impossible. For instance, the first dust model of the 1930's was made of small iron particles (e.g. Schoenberg & Jung, 1937; Greenstein, 1938), in part because Mie computations for small metal spheres were easier on paper than for large dielectrics (van de Hulst, 1986). The first dust radiative transfer numerical computations were performed in the early 1970's, using iterative methods (Mathis, 1970) and Monte-Carlo methods (Mattila, 1970).

**The development of modern detectors** solved the issues of photography: (i) non-linear response; (ii) restricted dynamic range; (iii) low detection efficiency; (iv) reciprocity failure; and (v) adjacency effects (Boksenberg, 1982). The first *Charge-Coupled Device* (CCD) was invented in 1969 at Bell laboratories (Amelio et al., 1970). In the IR, photomultipliers and bolometers were developed in the 1930's, and found important military applications during World War II and later: night vision and guiding rockets (Rogalski, 2012, for an extensive review). The first IR thermal detector had in fact been built 150 years earlier by Sir William HERSCHEL, in 1800. He used a prism to split the Sun light over several thermometers (cf. Fig. II.1.b; Rogalski, 2012). He found

<footnote n="2">2. A daguerreotype is the capture of an image directly on a chemically-treated metal plate, without the recourse to a negative.</footnote>
<figure><img src="image_159.png" /></figure>

that the highest temperature was beyond the red, that is in the infrared 3 (Fig. II.1.c).

**The possibility to send airborne and space observatories** opened the spectral windows where the atmosphere is opaque (cf. Sect. II.1.1.1). The interest to send telescopes in space was first advocated for by Lyman SPITZER Jr., in 1946 (Spitzer, 1946). The first successful space telescope was launched in 1968. It was the *Orbiting Astronomical Observatory (OAO)*, operating in the UV (Code et al., 1970). The first airborne observatory was the balloon experiment *Stratoscope I*, operating in the IR, in 1958.

The next technological innovation that will revolutionize our field is difficult to predict. We can however note that quantum computers should permit *ab initio* calculations of the properties and evolution of complex molecules and solids, in the near future. If this is true, it should allow us to precisely characterize the dust composition in different environments.

## II.1.1 The Challenges of Observing Interstellar Regions

### II.1.1.1 Limitations Due to the Atmosphere

The atmosphere of Earth is transparent in only a few spectral windows. The bottom panel of Fig. II.2 shows its absorbance as a function of wavelength. We see in particular that the UV and FIR ranges are completely opaque, and thus inaccessible from the ground. This is the reason why the first evidences for ISD came from the extinction of starlight in the visible range. The last forty years have seen the development of space and airborne observatories in the UV and IR windows, providing us with a panchromatic view of ISD properties (top panel of Fig. II.2).

### II.1.1.2 Historical Ground-Based Observatories

**The meter-class visible telescopes.** The first discoveries about ISD were performed in the visible domain, by understanding the extinction toward stars. Robert TRUMPLER's seminal paper (Trumpler, 1930) was based on observations made at the Lick observatory, near San Jose, as well as Edward BARNARD's most famous observations (Barnard, 1899, 1919). Such telescopes, all across the world, were the main source of empirical constraints on dust, until the end of the 1970's.

**The first MIR telescopes.** Ground-based observations, at high altitude, in dry regions of the globe such as the Mauna Kea, are possible in the MIR (cf. Fig. II.2). IR astronomy really started in the 1960's (cf. Walker, 2000, for an historical review). The first IR surveys of the northern and southern skies were performed by Neugebauer & Leighton (1968) and Price (1968), at 2 µm. A generation of 2-to-3-m MIR telescopes were commissioned at the end of the 1970's, such as the *Wyoming InfraRed Observatory* (WIRO; 1977; $\emptyset$ = 2.3 m), the *United Kingdom InfraRed Telescope* (UKIRT; 1978; $\emptyset$ = 3.8 m) and the *InfraRed Telescope Facility* (IRTF; 1979; $\emptyset$ = 3 m). Current large telescopes such as Subaru (Mauna Kea; $\emptyset$ = 8.2 m) or the *Very Large Telescope* (VLT; Paranal; $\emptyset$ = 8.2 m) operate in the visible-to-MIR range.

**The submillimeter observatories.** On the other side of the FIR atmospheric absorption, the submillimeter domain can be observed from the ground in dry conditions. The first ground-based submillimeter observatories appeared in 1990's. Among them are: the *Caltech Submillimeter Observatory* (CSO; Mauna Kea; $\emptyset$ = 10 m; 1986-2015), the *James Clerk Maxwell Telescope* (JCMT; Mauna Kea; $\emptyset$ = 15 m; 1987), the *Atacama Pathfinder Experiment* (APEX; Atacama desert; $\emptyset$ = 12 m; 2004) and the *Atacama Large Millimeter/submillimeter Array* (ALMA; Atacama desert; interferometer; 2011).

<figure><img src="image_160.png" /></figure>
<figure><img src="image_161.png" /></figure>

**FIGURE II.2 – Absorbance of Earth's atmosphere.** The bottom panel shows the absorbance of the atmosphere, in blue. Most of the absorption is due to H₂O, with some contribution by N₂, O₂, O₃, N₂O, CH₄ and CO₂. We have displayed in grey, in the background, the typical SED of a galaxy, for reference. We see that the UV and FIR windows are totally opaque from the ground (cf. Table A.4 for the denomination of the different spectral windows). Consequently, observations in these spectral windows can only be achieved from space, or above the troposphere (stratospheric airplane or balloon). Licensed under CC BY-SA 4.0.

### II.1.1.3 Airborne Observatories

**Balloons.** Stratospheric balloons can reach altitudes of ≈ 40 km, well above most water vapor absorption. They can observe for several days continuously, but landing is hazardous (Fig. II.3.a). The first IR balloon was launched from Johns Hopkins in 1959, and a balloon sent by the Goddard Institute of Space Sciences mapped the sky at 100 µm in 1966 (Walker, 2000). During the past two decades, several balloons provided observations of the dust continuum intensity and polarization in different regions of the sky, including: the *Balloon-borne Large Aperture Submillimeter Telescope* (BLAST);

<footnote n="3">3. The photosphere of the Sun is indeed a T ≈ 5800 K black body, peaking at λmax ≈ 0.88 µm. This is the first SED in history.</footnote>