Preface ....................................................... xix
1 The dielectric permeability .................................. 1
1.1 Maxwell's equations ..................................... 1
1.1.1 Electric field and magnetic induction ............ 1
1.1.2 Electric polarization of the medium .............. 2
1.1.3 The dependence of the dielectric permeability
on direction and frequency ....................... 3
1.1.4 The physical meaning of the electric
susceptibility Χ ................................. 4
1.1.5 Magnetic polarization of the medium .............. 6
1.1.6 The magnetic susceptibility ...................... 7
1.1.7 Dielectrics and metals ........................... 7
1.1.8 Free charges and polarization charges ............ 8
1.1.9 The field equations .............................. 9
1.2 Waves in a dielectric medium ........................... 10
1.2.1 The wave equation ............................... 10
1.2.2 The wavenumber .................................. 11
1.2.3 The optical constant or refractive index ........ 12
1.2.4 Energy dissipation of a grain in a variable
field ........................................... 13
1.3 The harmonic oscillator ................................ 15
1.3.1 The Lorentz model ............................... 15
1.3.2 Free oscillations ............................... 16
1.3.3 The general solution to the oscillator
equation ........................................ 17
1.3.4 Dissipation of energy in a forced oscillation ... 18
1.3.5 Dissipation of energy in a free oscillation ..... 19
1.3.6 The plasma frequency ............................ 20
1.3.7 Dispersion relation of the dielectric
permeability .................................... 20
1.4 The harmonic oscillator and light ...................... 22
1.4.1 Attenuation and refraction of light ............. 23
1.4.2 Retarded potentials of a moving charge .......... 24
1.4.3 Emission of an harmonic oscillator .............. 26
1.4.4 Radiation of higher order ....................... 27
1.4.5 Radiation damping ............................... 28
1.4.6 The cross section of an harmonic oscillator ..... 29
1.4.7 The oscillator strength ......................... 30
1.4.8 The natural linewidth ........................... 31
1.5 Waves in a conducting medium ........................... 32
1.5.1 The dielectric permeability of a conductor ...... 33
1.5.2 Conductivity and the Drude profile .............. 34
1.5.3 Electromagnetic waves in a plasma with
a magnetic field ................................ 36
1.5.4 Group velocity of electromagnetic waves in
a plasma ........................................ 37
1.6 Polarization through orientation ....................... 38
1.6.1 Polarization in a constant field ................ 38
1.6.2 Polarization in a time-variable field ........... 39
1.6.3 Relaxation after switching off the field ........ 40
1.6.4 The dielectric permeability in Debye
relaxation ...................................... 41
2 How to evaluate grain cross sections ........................ 44
2.1 Defining cross sections ................................ 44
2.1.1 Cross section for scattering, absorption and
extinction ...................................... 44
2.1.2 Cross section for radiation pressure ............ 46
2.1.3 Efficiencies, mass and volume coefficients ...... 47
2.2 The optical theorem .................................... 47
2.2.1 The intensity of forward scattered light ........ 47
2.2.2 The refractive index of a dusty medium .......... 50
2.3 Mie theory for a sphere ................................ 51
2.3.1 The generating function ......................... 52
2.3.2 Separation of variables ......................... 52
2.3.3 Series expansion of waves ....................... 54
2.3.4 Expansion coefficients .......................... 54
2.3.5 Scattered and absorbed power .................... 56
2.3.6 Absorption and scattering efficiencies .......... 57
2.4 Polarization and scattering ............................ 57
2.4.1 The amplitude scattering matrix ................. 57
2.4.2 Angle-dependence of scattering .................. 58
2.4.3 The polarization ellipse ........................ 60
2.4.4 Stokes parameters ............................... 61
2.4.5 Stokes parameters of scattered light for
a sphere ........................................ 62
2.5 The Kramers-Kronig relations ........................... 64
2.5.1 Mathematical formulation of the relations ....... 64
2.5.2 The electric susceptibility and causality ....... 66
2.5.3 The Kramers-Kronig relation for the dielectric
permeability .................................... 67
2.5.4 Extension to metals ............................. 67
2.5.5 Dispersion of the magnetic susceptibility ....... 68
2.5.6 Three corollaries of the KK relation ............ 69
2.6 Composite grains ....................................... 71
2.6.1 Effective medium theories ....................... 72
2.6.2 Garnett's mixing rule ........................... 73
2.6.3 The mixing rule of Bruggeman .................... 74
2.6.4 Composition of grains in protostellar cores ..... 74
2.6.5 How size, ice and porosity change
the absorption coefficient ...................... 76
3 Very small and very big particles .......................... 80
3.1 Tiny spheres ........................................... 80
3.1.1 When is a particle in the Rayleigh limit? ....... 80
3.1.2 Efficiencies of small spheres from Mie theory ... 81
3.1.3 A dielectric sphere in a constant electric
field ........................................... 82
3.1.4 Scattering and absorption in the electrostatic
approximation ................................... 84
3.1.5 Polarization and angle-dependent scattering ..... 85
3.1.6 Small-size effects beyond Mie theory ............ 86
3.2 A small metallic sphere in a magnetic field ............ 87
3.2.1 Slowly varying field ............................ 87
3.2.2 The magnetic polarizability ..................... 89
3.2.3 The penetration depth ........................... 89
3.2.4 Limiting values of the magnetic
polarizability .................................. 90
3.3 Tiny ellipsoids ........................................ 90
3.3.1 Elliptical coordinates .......................... 91
3.3.2 An ellipsoid in a constant electric field ....... 92
3.3.3 Cross section and shape factor .................. 93
3.3.4 Randomly oriented ellipsoids .................... 95
3.3.5 Pancakes and cigars ............................. 95
3.3.6 Rotation about the axis of greatest moment of
inertia ......................................... 97
3.4 The fields inside a dielectric particle ................ 99
3.4.1 Internal field and depolarization field ......... 99
3.4.2 Depolarization field and the distribution of
surface charges ................................ 100
3.4.3 The local field at an atom ..................... 101
3.4.4 The Clausius-Mossotti relation ................. 101
3.5 Very large particles .................................. 103
3.5.1 Babinet's theorem .............................. 103
3.5.2 Reflection and transmission at a plane
surface ........................................ 104
3.5.3 Huygens' principle ............................. 106
3.5.4 Fresnel zones and a check on Huygens
'principle ..................................... 109
3.5.5 The reciprocity theorem ........................ 111
3.5.6 Diffraction by a circular hole or a sphere ..... 111
3.5.7 Diffraction behind a half-plane ................ 113
3.5.8 Particles of small refractive index ............ 116
3.5.9 X-ray scattering ............................... 117
4 Case studies of Mie calculus ............................... 119
4.1 Efficiencies of bare spheres .......................... 119
4.1.1 Pure scattering ................................ 119
4.1.2 A weak absorber ................................ 120
4.1.3 A strong absorber .............................. 122
4.1.4 A metal sphere ................................. 123
4.1.5 Efficiency versus cross section and volume
coefficient .................................... 123
4.1.6 The atmosphere of the Earth .................... 126
4.2 Scattering by bare spheres ............................ 127
4.2.1 The scattering diagram ......................... 127
4.2.2 The polarization of scattered light ............ 128
4.2.3 The intensity of scattered light in
a reflection nebula ............................ 131
4.3 Coated spheres ........................................ 132
4.4 Surface modes in small grains ......................... 133
4.5 Efficiencies of idealized dielectrics and metals ...... 136
4.5.1 Dielectric sphere consisting of identical
harmonic oscillators ........................... 136
4.5.2 Dielectric sphere with Debye relaxation ........ 138
4.5.3 Magnetic and electric dipole absorption of
small metal spheres ............................ 139
4.5.4 Efficiencies for Drude profiles ................ 141
4.5.5 Elongated metallic particles ................... 142
5 Particle statistics ........................................ 145
5.1 Boltzmann statistics .................................. 145
5.1.1 The probability of an arbitrary energy
distribution ................................... 145
5.1.2 The distribution of maximum probability ........ 146
5.1.3 Partition function and population of energy
cells .......................................... 147
5.1.4 The mean energy of harmonic oscillators ........ 149
5.1.5 The Maxwellian velocity distribution ........... 149
5.2 Quantum statistics .................................... 151
5.2.1 The unit cell h3 of the phase space ............ 151
5.2.2 Bosons and fermions ............................ 152
5.2.3 Bose statistics ................................ 154
5.2.4 Bose statistics for photons .................... 156
5.2.5 Fermi statistics ............................... 157
5.2.6 Ionization equilibrium and the Saha equation ... 158
5.3 Thermodynamics ........................................ 160
5.3.1 The ergodic hypothesis ......................... 160
5.3.2 Definition of entropy and temperature .......... 162
5.3.3 The canonical distribution ..................... 163
5.3.4 Thermodynamic relations for a gas .............. 164
5.3.5 Equilibrium conditions of the state
functions ...................................... 166
5.3.6 Specific heat of a gas ......................... 168
5.3.7 The work done by magnetization ................. 168
5.3.8 Susceptibility and specific heat of magnetic
substances ..................................... 169
5.4 Blackbody radiation ................................... 170
5.4.1 The Planck function ............................ 170
5.4.2 Low-and high-frequency limit ................... 171
5.4.3 Wien's displacement law and the Stefan-
Boltzmann law .................................. 172
5.4.4 The Planck function and harmonic oscillators ... 173
6 The radiative transition probability ....................... 175
6.1 A charged particle in an electromagnetic field ........ 175
6.1.1 The classical Hamiltonian ...................... 175
6.1.2 The Hamiltonian of an electron in an
electromagnetic field .......................... 176
6.1.3 The Hamilton operator in quantum mechanics ..... 177
6.1.4 The dipole moment in quantum mechanics ......... 179
6.1.5 The quantized harmonic oscillator .............. 179
6.2 Small perturbations ................................... 181
6.2.1 The perturbation energy ........................ 181
6.2.2 The transition probability ..................... 181
6.2.3 Transition probability for a time-variable
perturbation ................................... 182
6.3 The Einstein coefficients A and В ..................... 183
6.3.1 Induced and spontaneous transitions ............ 183
6.3.2 Selection rules and polarization rules ......... 186
6.3.3 Quantization of the electromagnetic field ...... 186
6.3.4 Quantum-mechanical derivation of A and В ....... 188
6.4 Potential wells and tunneling ......................... 192
6.4.1 Wavefunction of a particle in a constant
potential ...................................... 192
6.4.2 Potential walls and Fermi energy ............... 192
6.4.3 Rectangular potential barriers ................. 194
6.4.4 The double potential well ...................... 198
7 Structure and composition of dust .......................... 201
7.1 Crystal structure ..................................... 201
7.1.1 Translational symmetry ......................... 201
7.1.2 Lattice types .................................. 203
7.1.3 The reciprocal lattice ......................... 207
7.2 Binding in crystals ................................... 207
7.2.1 Covalent bonding ............................... 208
7.2.2 Ionic bonding .................................. 209
7.2.3 Metals ......................................... 211
7.2.4 van der Waals forces and hydrogen bridges ...... 213
7.3 Reddening by interstellar grains ...................... 214
7.3.1 Stellar photometry ............................. 214
7.3.2 The interstellar extinction curve .............. 216
7.3.3 Two-color diagrams ............................. 219
7.3.4 Spectral indices ............................... 220
7.3.5 The mass absorption coefficient .............. 222
7.4 Carbonaceous grains and silicate grains ............... 224
7.4.1 Origin of the two major dust constituents ...... 224
7.4.2 The bonding in carbon .......................... 225
7.4.3 Carbon compounds ............................... 227
7.4.4 Silicates ...................................... 232
7.4.5 A standard set of optical constants ............ 233
7.5 Grain sizes and optical constants ..................... 234
7.5.1 The size distribution .......................... 234
7.5.2 Collisional fragmentation ...................... 236
8 Dust radiation ............................................. 239
8.1 Kirchhoff's law ....................................... 239
8.1.1 The emissivity of dust ......................... 239
8.1.2 Thermal emission of grains ..................... 240
8.1.3 Absorption and emission in thermal
equilibrium .................................... 241
8.1.4 Equipartition of energy ........................ 242
8.2 The temperature of big grains ......................... 243
8.2.1 The energy equation ............................ 243
8.2.2 Approximate absorption efficiency at infrared
wavelengths .................................... 243
8.2.3 Temperature estimates .......................... 245
8.2.4 Relation between grain size and grain
temperature .................................... 247
8.2.5 Temperature of dust grains near a star ......... 248
8.2.6 Dust temperatures from observations ............ 249
8.3 The emission spectrum of big grains ................... 251
8.3.1 Constant temperature and low optical depth ..... 251
8.3.2 Constant temperature and arbitrary optical
depth .......................................... 253
8.4 Calorific properties of solids ........................ 254
8.4.1 Normal coordinates ............................. 254
8.4.2 Internal energy of a grain ..................... 256
8.4.3 Standing waves in a crystal .................... 257
8.4.4 The density of vibrational modes in
a crystal ...................................... 258
8.4.5 Specific heat .................................. 259
8.4.6 Two-dimensional lattices ....................... 261
8.5 Temperature fluctuations of very small grains ......... 262
8.5.1 The probability density P(T) ................... 263
8.5.2 The transition matrix .......................... 263
8.5.3 Practical considerations ....................... 265
8.5.4 The stochastic time evolution of grain
temperature .................................... 266
8.6 The emission spectrum of very small grains ............ 268
8.6.1 Small and moderate fluctuations ................ 268
8.6.2 Strong fluctuations ............................ 270
8.6.3 Temperature fluctuations and flux ratios ....... 272
9 Dust and its environment ................................... 275
9.1 Grain surfaces ........................................ 275
9.1.1 Gas accretion on grains ........................ 275
9.1.2 Physical adsorption and chemisorption .......... 276
9.1.3 The sticking probability ....................... 279
9.1.4 Thermal hopping, evaporation and reactions
with activation barrier ........................ 281
9.1.5 Tunneling between surface sites ................ 283
9.1.6 Scanning time .................................. 284
9.2 Grain charge .......................................... 285
9.2.1 Charge equilibrium in the absence of a UV
radiation field ................................ 285
9.2.2 The photoelectric effect ....................... 286
9.3 Grain motion .......................................... 289
9.3.1 Random walk .................................... 289
9.3.2 The drag on a grain subjected to a constant
outer force .................................... 289
9.3.3 Brownian motion of a grain ..................... 292
9.3.4 The disorder time .............................. 293
9.3.5 Laminar and turbulent friction ................. 295
9.3.6 A falling rain drop ............................ 296
9.3.7 The Poynting-Robertson effect .................. 297
9.4 Grain destruction ..................................... 298
9.4.1 Mass balance in the Milky Way .................. 298
9.4.2 Destruction processes .......................... 299
9.5 Grain formation ....................................... 301
9.5.1 Evaporation temperature of dust ................ 301
9.5.2 Vapor pressure of small grains ................. 304
9.5.3 Critical saturation ............................ 305
9.5.4 Equations for time-dependent homogeneous
nucleation ..................................... 307
9.5.5 Equilibrium distribution and steady-state
nucleation ..................................... 308
9.5.6 Solutions to time-dependent homogeneous
nucleation ..................................... 311
9.5.7 Similarity relations ........................... 316
10 Polarization ............................................... 319
10.1 Efficiency of infinite cylinders ................. 319
10.1.1 Normal incidence and picket fence alignment .... 319
10.1.2 Oblique incidence .............................. 322
10.1.3 Rotating cylinders ............................. 322
10.1.4 Absorption efficiency as a function of
wavelength ..................................... 325
10.2 Linear polarization through extinction ................ 327
10.2.1 Effective optical depth and degree of
polarization p(λ) .............................. 327
10.2.2 The Serkowski curve ............................ 329
10.2.3 Polarization p(λ) of infinite cylinders ........ 331
10.2.4 Polarization p(λ) of ellipsoids in the
Rayleigh limit ................................. 334
10.2.5 Polarization p(λ) of spheroids at optical
wavelengths .................................... 337
10.2.6 Polarization and reddening ..................... 338
10.3 Polarized emission .................................... 339
10.3.1 The wavelength dependence of polarized
emission for cylinders ......................... 340
10.3.2 Infrared emission of spheroids ................. 340
10.3.3 Polarized emission versus polarized
extinction ..................................... 341
10.4 Circular polarization ................................. 342
10.4.1 The phase shift induced by grains .............. 343
10.4.2 The wavelength dependence of circular
polarization ................................... 344
11 Grain alignment ............................................ 347
11.1 Grain rotation ........................................ 347
11.1.1 Euler's equations for a rotating body .......... 347
11.1.2 Symmetric tops ................................. 349
11.1.3 Atomic magnet in a magnetic field .............. 351
11.1.4 Rotational Brownian motion ..................... 351
11.1.5 Suprathermal rotation .......................... 353
11.2 Magnetic dissipation .................................. 355
11.2.1 Diamagnetism ................................... 355
11.2.2 Paramagnetism .................................. 355
11.2.3 Ferromagnetism ................................. 357
11.2.4 The magnetization of iron above and below
the Curie point ................................ 358
11.2.5 Paramagnetic dissipation: spin-spin and
spin-lattice relaxation ........................ 359
11.2.6 The magnetic susceptibility for spin-lattice
relaxation ..................................... 360
11.2.7 The magnetic susceptibility in spin-spin
relaxation ..................................... 362
11.3 Magnetic alignment .................................... 364
11.3.1 A rotating dipole in a magnetic field .......... 365
11.3.2 Timescales for alignment and disorder .......... 367
11.3.3 Super-paramagnetism ............................ 368
11.3.4 Ferromagnetic relaxation ....................... 369
11.3.5 Alignment of angular momentum with the axis
of greatest inertia ............................ 371
11.3.6 Mechanical and magnetic damping ................ 372
11.4 Non-magnetic alignment ................................ 373
11.4.1 Gas streaming .................................. 373
11.4.2 Anisotropic illumination ....................... 375
12 PAHs and spectral features of dust ......................... 377
12.1 Thermodynamics of PAHs ................................ 377
12.1.1 What are PAHs? ................................. 377
12.1.2 Microcanonic emission of PAHs .................. 378
12.1.3 The vibrational modes of anthracene ............ 379
12.1.4 Microcanonic versus thermal level population ... 381
12.1.5 Does an ensemble of PAHs have a temperature? ... 382
12.2 PAH emission .......................................... 384
12.2.1 Photoexcitation of PAHs ........................ 384
12.2.2 Cutoff wavelength for electronic excitation .... 385
12.2.3 Photo-destruction and ionization ............... 386
12.2.4 Cross sections and line profiles of PAHs ....... 387
12.3 Big grains and ices ................................... 388
12.3.1 The silicate features and the band at 3.4
μm ............................................. 389
12.3.2 Icy grain mantles .............................. 389
12.4 An overall dust model ................................. 390
12.4.1 The three dust components ...................... 392
12.4.2 Extinction coefficient in the diffuse medium ... 395
12.4.3 Extinction coefficient in protostellar cores ... 395
13 Radiative transport ........................................ 396
13.1 Basic transfer relations .............................. 396
13.1.1 Radiative intensity and flux ................... 396
13.1.2 The transfer equation and its formal
solution ....................................... 398
13.1.3 The brightness temperature ..................... 400
13.1.4 The main-beam-brightness temperature of
a telescope .................................... 401
13.2 Spherical clouds ...................................... 402
13.2.1 Moment equations for spheres ................... 403
13.2.2 Frequency averages ............................. 404
13.2.3 Differential equations for the intensity ....... 405
13.2.4 Integral equations for the intensity ........... 407
13.2.5 Practical hints ................................ 407
13.3 Passive disks ......................................... 409
13.3.1 Radiative transfer in a plane parallel layer ... 409
13.3.2 The grazing angle in an inflated disk .......... 414
13.4 Galactic nuclei ....................................... 415
13.4.1 Hot spots in a spherical stellar cluster ....... 415
13.4.2 Low and high luminosity stars .................. 416
13.5 Line radiation ........................................ 418
13.5.1 Absorption coefficient and absorption
profile ........................................ 418
13.5.2 The excitation temperature of a line ........... 419
13.5.3 Radiative transfer in lines .................... 420
14 Diffuse matter in the Milky Way ............................ 425
14.1 Overview of the Milky Way ............................. 425
14.1.1 Global parameters .............................. 425
14.1.2 The relevance of dust .......................... 426
14.2 Molecular clouds ...................................... 427
14.2.1 The CO molecule ................................ 428
14.2.2 Population of levels in CO ..................... 431
14.2.3 Molecular hydrogen ............................. 435
14.2.4 Formation of molecular hydrogen on dust
surfaces ....................................... 435
14.3 Clouds of atomic hydrogen ............................. 438
14.3.1 General properties of the diffuse gas .......... 438
14.3.2 The 21 cm line of atomic hydrogen .............. 439
14.3.3 How the hyperfine levels of atomic hydrogen
are excited .................................... 440
14.3.4 Gas density and temperature from the 21 cm
line ........................................... 443
14.3.5 The deuterium hyperfine line ................... 444
14.3.6 Electron density and magnetic field in
the diffuse gas ................................ 446
14.4 HII regions ........................................... 448
14.4.1 Ionization and recombination ................... 448
14.4.2 Dust-free HII regions .......................... 450
14.4.3 Dusty HII regions .............................. 453
14.4.4 Bremsstrahlung ................................. 455
14.4.5 Recombination lines ............................ 456
14.5 Mass estimates of interstellar clouds ................. 457
14.5.1 From optically thin CO lines ................... 457
14.5.2 From the CO luminosity ......................... 458
14.5.3 From dust emission ............................. 459
15 Stars and their formation .................................. 461
15.1 Stars on and beyond the main sequence ................. 461
15.1.1 Nuclear burning and the creation of elements ... 461
15.1.2 The binding energy of an atomic nucleus ........ 463
15.1.3 Hydrogen burning ............................... 465
15.1.4 The 3a process ................................. 467
15.1.5 Lifetime and luminosity of stars ............... 469
15.1.6 The initial mass function ...................... 470
15.2 Clouds near gravitational equilibrium ................. 471
15.2.1 Virialized clouds .............................. 471
15.2.2 Isothermal cloud in pressure equilibrium ....... 474
15.2.3 Structure and stability of Ebert-Bonnor
spheres ........................................ 475
15.2.4 Free-fall of a gas ball ........................ 479
15.2.5 The critical mass for gravitational
instability .................................... 480
15.2.6 Implications of the Jeans criterion ............ 482
15.2.7 Magnetic fields and ambipolar diffusion ........ 484
15.3 Gravitational collapse ................................ 486
15.3.1 The presolar nebula ............................ 486
15.3.2 Hydrodynamic collapse simulations .............. 487
15.3.3 Similarity solutions of collapse ............... 491
15.4 Disks ................................................. 494
15.4.1 Viscous laminar flows .......................... 494
15.4.2 Dynamical equations of the thin accretion
disk ........................................... 497
15.4.3 The Kepler disk ................................ 498
15.4.4 Why a star accretes from a disk ................ 499
15.4.5 The stationary accretion disk .................. 501
15.4.6 Thea-disk ...................................... 501
15.4.7 Disk heating by viscosity ...................... 503
16 Emission from young stars .................................. 505
16.1 The earliest stages of star formation ................. 505
16.1.1 Globules ....................................... 505
16.1.2 Isothermal gravitationally-bound clumps ........ 506
16.2 The collapse phase .................................... 508
16.2.1 The density structure of a protostar ........... 508
16.2.2 Dust emission from a solar-type protostar ...... 513
16.2.3 Kinematics of protostellar collapse ............ 515
16.3 Accretion disks ....................................... 518
16.3.1 A flat blackbody disk .......................... 518
16.3.2 A flat non-blackbody disk ...................... 521
16.3.3 Radiative transfer in an inflated disk ......... 522
16.4 Reflection nebulae .................................... 524
16.5 Cold and warm dust in galaxies ........................ 526
16.6 Starburst nuclei ...................................... 531
16.6.1 Repetitive bursts of star formation ............ 531
16.6.2 Dust emission from starburst nuclei ............ 535
Appendix A Mathematical formulae .............................. 539
Appendix В List of symbols .................................... 545
References .................................................... 549
Index.......................................................... 552
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