THE PROJECT PHYSICS COURSE: MOTION IN THE HEAVENS
1 The dielectric permeability 1 11 Maxwell’s equations 1 111 Electric field and magnetic induction 1 112 Electric polarization of the medium 2 113 The dependence of the dielectric permeability on direction and frequency 3 114 The physical meaning of the electric susceptibility ? 4 115 Magnetic polarization of the medium 6 116 The magnetic susceptibility 7 117 Dielectrics and metals 7 118 Free charges and polarization charges 8 119 The field equations 9 12 Waves in a dielectric medium 10 121 The wave equation 10 122 The wavenumber 11 123 The optical constant or refractive index 12 124 Energy dissipation of a grain in a variable field 13 13 The harmonic oscillator 15 131 The Lorentz model 15 132 Free oscillations 16 133 The general solution to the oscillator equation 17 134 Dissipation of energy in a forced oscillation 18 135 Dissipation of energy in a free oscillation 19 136 The plasma frequency 20 137 Dispersion relation of the dielectric permeability 20 14 The harmonic oscillator and light 22 141 Attenuation and refraction of light 23 142 Retarded potentials of a moving charge 24 143 Emission of an harmonic oscillator 26 144 Radiation of higher order 27 145 Radiation damping 28 146 The cross section of an harmonic oscillator 29 147 The oscillator strength 30 148 The natural linewidth 31 15 Waves in a conducting medium 32 151 The dielectric permeability of a conductor 33 152 Conductivity and the Drude profile 34 153 Electromagnetic waves in a plasma with a magnetic field 36 154 Group velocity of electromagnetic waves in a plasma 37 16 Polarization through orientation 38 161 Polarization in a constant field 38 162 Polarization in a time-variable field 39 163 Relaxation after switching off the field 40 164 The dielectric permeability in Debye relaxation 41 2 How to evaluate grain cross sections 44 21 Defining cross sections 44 211 Cross section for scattering, absorption and extinction 44 212 Cross section for radiation pressure 46 213 Efficiencies, mass and volume coefficients 47 22 The optical theorem 47 221 The intensity of forward scattered light 47 222 The refractive index of a dusty medium 50 23 Mie theory for a sphere 51 231 The generating function 52 232 Separation of variables 52 233 Series expansion of waves 54 234 Expansion coefficients 54 235 Scattered and absorbed power 56 236 Absorption and scattering efficiencies 57 24 Polarization and scattering 57 241 The amplitude scattering matrix 57 242 Angle-dependence of scattering 58 243 The polarization ellipse 60 244 Stokes parameters 61 245 Stokes parameters of scattered light for a sphere 62 25 The Kramers–Kronig relations 64 251 Mathematical formulation of the relations 64 252 The electric susceptibility and causality 66 253 The Kramers–Kronig relation for the dielectric permeability 67 254 Extension to metals 67 255 Dispersion of the magnetic susceptibility 68 256 Three corollaries of the KK relation 69 26 Composite grains 71 261 Effective medium theories 72 262 Garnett’s mixing rule 73 263 The mixing rule of Bruggeman 74 264 Composition of grains in protostellar cores 74 265 How size, ice and porosity change the absorption coefficient 76 3 Very small and very big particles 80 31 Tiny spheres 80 311 When is a particle in the Rayleigh limit? 80 312 Efficiencies of small spheres from Mie theory 81 313 A dielectric sphere in a constant electric field 82 314 Scattering and absorption in the electrostatic approximation 84 315 Polarization and angle-dependent scattering 85 316 Small-size effects beyond Mie theory 86 32 A small metallic sphere in a magnetic field 87 321 Slowly varying field 87 322 The magnetic polarizability 89 323 The penetration depth 89 324 Limiting values of the magnetic polarizability 90 33 Tiny ellipsoids 90 331 Elliptical coordinates 91 332 An ellipsoid in a constant electric field 92 333 Cross section and shape factor 93 334 Randomly oriented ellipsoids 95 335 Pancakes and cigars 95 336 Rotation about the axis of greatest moment of inertia 97 34 The fields inside a dielectric particle 99 341 Internal field and depolarization field 99 342 Depolarization field and the distribution of surface charges 100 343 The local field at an atom 101 344 The Clausius–Mossotti relation 101 35 Very large particles 103 351 Babinet’s theorem 103 352 Reflection and transmission at a plane surface 104 353 Huygens’ principle 106 354 Fresnel zones and a check on Huygens’ principle 109 355 The reciprocity theorem 111 356 Diffraction by a circular hole or a sphere 111 357 Diffraction behind a half-plane 113 358 Particles of small refractive index 116 359 X-ray scattering 117 4 Case studies of Mie calculus 119 41 Efficiencies of bare spheres 119 411 Pure scattering 119 412 A weak absorber 120 413 A strong absorber 122 414 A metal sphere 123 415 Efficiency versus cross section and volume coefficient 123 416 The atmosphere of the Earth 126 42 Scattering by bare spheres 127 421 The scattering diagram 127 422 The polarization of scattered light 128 423 The intensity of scattered light in a reflection nebula 131 43 Coated spheres 132 44 Surface modes in small grains 133 45 Efficiencies of idealized dielectrics and metals 136 451 Dielectric sphere consisting of identical harmonic oscillators 136 452 Dielectric sphere with Debye relaxation 138 453 Magnetic and electric dipole absorption of small metal spheres 139 454 Efficiencies for Drude profiles 141 455 Elongated metallic particles 142 5 Particle statistics 145 51 Boltzmann statistics 145 511 The probability of an arbitrary energy distribution 145 512 The distribution of maximum probability 146 513 Partition function and population of energy cells 147 514 The mean energy of harmonic oscillators 149 515 The Maxwellian velocity distribution 149 52 Quantum statistics 151 521 The unit cell h3 of the phase space 151 522 Bosons and fermions 152 523 Bose statistics 154 524 Bose statistics for photons 156 525 Fermi statistics 157 526 Ionization equilibrium and the Saha equation 158 53 Thermodynamics 160 531 The ergodic hypothesis 160 532 Definition of entropy and temperature 162 533 The canonical distribution 163 534 Thermodynamic relations for a gas 164 535 Equilibrium conditions of the state functions 166 536 Specific heat of a gas 168 537 The work done by magnetization 168 538 Susceptibility and specific heat of magnetic substances 169 54 Blackbody radiation 170 541 The Planck function 170 542 Low- and high-frequency limit 171 543 Wien’s displacement law and the Stefan–Boltzmann law 172 544 The Planck function and harmonic oscillators 173 6 The radiative transition probability 175 61 A charged particle in an electromagnetic field 175 611 The classical Hamiltonian 175 612 The Hamiltonian of an electron in an electromagnetic field 176 613 The Hamilton operator in quantum mechanics 177 614 The dipole moment in quantum mechanics 179 615 The quantized harmonic oscillator 179 62 Small perturbations 181 621 The perturbation energy 181 622 The transition probability 181 623 Transition probability for a time-variable perturbation 182 63 The Einstein coefficients A and B 183 631 Induced and spontaneous transitions 183 632 Selection rules and polarization rules 186 633 Quantization of the electromagnetic field 186 634 Quantum-mechanical derivation of A and B 188 64 Potential wells and tunneling 192 641 Wavefunction of a particle in a constant potential 192 642 Potential walls and Fermi energy 192 643 Rectangular potential barriers 194 644 The double potential well 198 7 Structure and composition of dust 201 71 Crystal structure 201 711 Translational symmetry 201 712 Lattice types 203 713 The reciprocal lattice 207 72 Binding in crystals 207 721 Covalent bonding 208 722 Ionic bonding 209 723 Metals 211 724 van der Waals forces and hydrogen bridges 213 73 Reddening by interstellar grains 214 731 Stellar photometry 214 732 The interstellar extinction curve 216 733 Two-color diagrams 219 734 Spectral indices 220 735 The mass absorption coefficient 222 74 Carbonaceous grains and silicate grains 224 741 Origin of the two major dust constituents 224 742 The bonding in carbon 225 743 Carbon compounds 227 744 Silicates 232 745 A standard set of optical constants 233 75 Grain sizes and optical constants 234 751 The size distribution 234 752 Collisional fragmentation 236 8 Dust radiation 239 81 Kirchhoff’s law 239 811 The emissivity of dust 239 812 Thermal emission of grains 240 813 Absorption and emission in thermal equilibrium 241 814 Equipartition of energy 242 82 The temperature of big grains 243 821 The energy equation 243 822 Approximate absorption efficiency at infrared wavelengths 243 823 Temperature estimates 245 824 Relation between grain size and grain temperature 247 825 Temperature of dust grains near a star 248 826 Dust temperatures from observations 249 83 The emission spectrum of big grains 251 831 Constant temperature and low optical depth 251 832 Constant temperature and arbitrary optical depth 253 84 Calorific properties of solids 254 841 Normal coordinates 254 842 Internal energy of a grain 256 843 Standing waves in a crystal 257 844 The density of vibrational modes in a crystal 258 845 Specific heat 259 846 Two-dimensional lattices 261 85 Temperature fluctuations of very small grains 262 851 The probability density P(T ) 263 852 The transition matrix 263 853 Practical considerations 265 854 The stochastic time evolution of grain temperature 266 86 The emission spectrum of very small grains 268 861 Small and moderate fluctuations 268 862 Strong fluctuations 270 863 Temperature fluctuations and flux ratios 272 9 Dust and its environment 275 91 Grain surfaces 275 911 Gas accretion on grains 275 912 Physical adsorption and chemisorption 276 913 The sticking probability 279 914 Thermal hopping, evaporation and reactions with activation barrier 281 915 Tunneling between surface sites 283 916 Scanning time 284 92 Grain charge 285 921 Charge equilibrium in the absence of a UV radiation field 285 922 The photoelectric effect 286 93 Grain motion 289 931 Random walk 289 932 The drag on a grain subjected to a constant outer force 289 933 Brownian motion of a grain 292 934 The disorder time 293 935 Laminar and turbulent friction 295 936 A falling rain drop 296 937 The Poynting–Robertson effect 297 94 Grain destruction 298 941 Mass balance in the Milky Way 298 942 Destruction processes 299 95 Grain formation 301 951 Evaporation temperature of dust 301 952 Vapor pressure of small grains 304 953 Critical saturation 305 954 Equations for time-dependent homogeneous nucleation 307 955 Equilibrium distribution and steady-state nucleation 308 956 Solutions to time-dependent homogeneous nucleation 311 957 Similarity relations 316 10 Polarization 319 101 Efficiency of infinite cylinders 319 1011 Normal incidence and picket fence alignment 319 1012 Oblique incidence 322 1013 Rotating cylinders 322 1014 Absorption efficiency as a function of wavelength 325 102 Linear polarization through extinction 327 1021 Effective optical depth and degree of polarization p(?) 327 1022 The Serkowski curve 329 1023 Polarization p(?) of infinite cylinders 331 1024 Polarization p(?) of ellipsoids in the Rayleigh limit 334 1025 Polarization p(?) of spheroids at optical wavelengths 337 1026 Polarization and reddening 338 103 Polarized emission 339 1031 The wavelength dependence of polarized emission for cylinders 340 1032 Infrared emission of spheroids 340 1033 Polarized emission versus polarized extinction 341 104 Circular polarization 342 1041 The phase shift induced by grains 343 1042 The wavelength dependence of circular polarization 344 11 Grain alignment 347 111 Grain rotation 347 1111 Euler’s equations for a rotating body 347 1112 Symmetric tops 349 1113 Atomic magnet in a magnetic field 351 1114 Rotational Brownian motion 351 1115 Suprathermal rotation 353 112 Magnetic dissipation 355 1121 Diamagnetism 355 1122 Paramagnetism 355 1123 Ferromagnetism 357 1124 The magnetization of iron above and below the Curie point 358 1125 Paramagnetic dissipation: spin–spin and spin–lattice relaxation 359 1126 The magnetic susceptibility for spin–lattice relaxation 360 1127 The magnetic susceptibility in spin–spin relaxation 362 113 Magnetic alignment 364 1131 A rotating dipole in a magnetic field 365 1132 Timescales for alignment and disorder 367 1133 Super-paramagnetism 368 1134 Ferromagnetic relaxation 369 1135 Alignment of angular momentum with the axis of greatest inertia 371 1136 Mechanical and magnetic damping 372 114 Non-magnetic alignment 373 1141 Gas streaming 373 1142 Anisotropic illumination 375 12 PAHs and spectral features of dust 377 121 Thermodynamics of PAHs 377 1211 What are PAHs? 377 1212 Microcanonic emission of PAHs 378 1213 The vibrational modes of anthracene 379 1214 Microcanonic versus thermal level population 381 1215 Does an ensemble of PAHs have a temperature? 382 122 PAH emission 384 1221 Photoexcitation of PAHs 384 1222 Cutoff wavelength for electronic excitation 385 1223 Photo-destruction and ionization 386 1224 Cross sections and line profiles of PAHs 387 123 Big grains and ices 388 1231 The silicate features and the band at 34 ?m 389 1232 Icy grain mantles 389 124 An overall dust model 390 1241 The three dust components 392 1242 Extinction coefficient in the diffuse medium 395 1243 Extinction coefficient in protostellar cores 395 13 Radiative transport 396 131 Basic transfer relations 396 1311 Radiative intensity and flux 396 1312 The transfer equation and its formal solution 398 1313 The brightness temperature 400 1314 The main-beam-brightness temperature of a telescope 401 132 Spherical clouds 402 1321 Moment equations for spheres 403 1322 Frequency averages 404 1323 Differential equations for the intensity 405 1324 Integral equations for the intensity 407 1325 Practical hints 407 133 Passive disks 409 1331 Radiative transfer in a plane parallel layer 409 1332 The grazing angle in an inflated disk 414 134 Galactic nuclei 415 1341 Hot spots in a spherical stellar cluster 415 1342 Low and high luminosity stars 416 135 Line radiation 418 1351 Absorption coefficient and absorption profile 418 1352 The excitation temperature of a line 419 1353 Radiative transfer in lines 420 14 Diffuse matter in the Milky Way 425 141 Overview of the Milky Way 425 1411 Global parameters 425 1412 The relevance of dust 426 142 Molecular clouds 427 1421 The CO molecule 428 1422 Population of levels in CO 431 1423 Molecular hydrogen 435 1424 Formation of molecular hydrogen on dust surfaces 435 143 Clouds of atomic hydrogen 438 1431 General properties of the diffuse gas 438 1432 The 21 cm line of atomic hydrogen 439 1433 How the hyperfine levels of atomic hydrogen are excited 440 1434 Gas density and temperature from the 21 cm line 443 1435 The deuterium hyperfine line 444 1436 Electron density and magnetic field in the diffuse gas 446 144 HII regions 448 1441 Ionization and recombination 448 1442 Dust–free HII regions 450 1443 Dusty HII regions 453 1444 Bremsstrahlung 455 1445 Recombination lines 456 145 Mass estimates of interstellar clouds 457 1451 From optically thin CO lines 457 1452 From the CO luminosity 458 1453 From dust emission 459 15 Stars and their formation 461 151 Stars on and beyond the main sequence 461 1511 Nuclear burning and the creation of elements 461 1512 The binding energy of an atomic nucleus 463 1513 Hydrogen burning 465 1514 The 3? process 467 1515 Lifetime and luminosity of stars 469 1516 The initial mass function 470 152 Clouds near gravitational equilibrium 471 1521 Virialized clouds 471 1522 Isothermal cloud in pressure equilibrium 474 1523 Structure and stability of Ebert–Bonnor spheres 475 1524 Free-fall of a gas ball 479 1525 The critical mass for gravitational instability 480 1526 Implications of the Jeans criterion 482 1527 Magnetic fields and ambipolar diffusion 484 153 Gravitational collapse 486 1531 The presolar nebula 486 1532 Hydrodynamic collapse simulations 487 1533 Similarity solutions of collapse 491 154 Disks 494 1541 Viscous laminar flows 494 1542 Dynamical equations of the thin accretion disk 497 1543 The Kepler disk 498 1544 Why a star accretes from a disk 499 1545 The stationary accretion disk 501 1546 The ?-disk 501 1547 Disk heating by viscosity 503 16 Emission from young stars 505 161 The earliest stages of star formation 505 1611 Globules 505 1612 Isothermal gravitationally-bound clumps 506 162 The collapse phase 508 1621 The density structure of a protostar 508 1622 Dust emission from a solar-type protostar 513 1623 Kinematics of protostellar collapse 515 163 Accretion disks 518 1631 A flat blackbody disk 518 1632 A flat non-blackbody disk 521 1633 Radiative transfer in an inflated disk 522 164 Reflection nebulae 524 165 Cold and warm dust in galaxies 526 166 Starburst nuclei 531 1661 Repetitive bursts of star formation 531 1662 Dust emission from starburst nuclei 535