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Fundamentals of Astrophysics Lecture Notes for ASTR 402/500

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Fundamentals of Astrophysics Lecture Notes for ASTR 402/500 Fundamentals of Astrophysics Lecture Notes for ASTR 402/500 Jeremy S. Heyl September 8, 2006 2 Contents 1 Radiative Transfer 7 1.1 Introduction . 7 1.2 Flux . 7 1.3 Intensity . 7 1.3.1 Relation to the flux . 8 1.3.2 Spectra . 9 1.3.3 An Astronomical Aside: Magnitudes . 9 1.4 Energy Density . 10 1.4.1 A Physical Aside: What are the Intensity and Flux? . 11 1.5 Blackbody Radiation . 11 1.5.1 Thermodynamics . 12 1.5.2 Statistical Mechanics . 13 1.5.3 Blackbody Temperatures . 15 1.6 Radiative Transfer . 16 1.6.1 Emission . 16 1.6.2 Absorption . 17 1.6.3 The Radiative Transfer Equation . 17 1.7 Thermal Radiation . 19 1.7.1 Einstein Coefficients . 19 1.7.2 A Physical Aside: What is deep about the Einstein coef- ficients? . 20 1.8 Calculating the Emission and Absoprtion Coefficients . 21 1.8.1 LTE . 21 1.8.2 Non-Thermal Emission . 22 1.9 Scattering . 22 1.9.1 Random Walks . 23 1.9.2 Combined Scattering and Absorption . 24 1.10 Radiative Diffusion . 25 1.10.1 Rosseland Approximation . 25 1.10.2 Eddington Approximation . 26 3 4 CONTENTS 2 Basic Theory of Radiation Fields 31 2.1 Maxwell’s Equations . 31 2.1.1 Waves . 33 2.1.2 The Spectrum . 34 2.1.3 An Mathematical Aside: Proving Parseval’s Theorem . 35 2.2 Polarization and Stokes Parameters . 36 2.3 Electromagnetic Potentials . 38 3 Radiation from Moving Charges 43 3.1 Retarded Potentials . 43 3.1.1 The Fields . 44 3.1.2 Non-relativistic particles . 45 3.1.3 How about relativistic particles? . 46 3.1.4 Radiation from Systems of Particles . 48 3.1.5 A Physical Aside: Multipole Radiation . 49 3.2 Thomson Scattering . 49 3.3 Radiation Reaction . 51 3.3.1 Radiation from Harmonically Bound Particles . 51 3.3.2 Driven Harmonic Oscillator . 52 4 Special Relativity 57 4.1 Back to Maxwell’s Equations . 57 4.2 Lorentz Transformations . 57 4.2.1 Length Contraction . 58 4.2.2 Time Dilation . 58 4.2.3 Adding velocities . 58 4.2.4 Doppler Effect . 59 4.3 Four-Vectors . 59 4.4 Tensors . 62 4.5 Transformation of Radiative Transfer . 64 5 Bremmstrahlung 69 5.1 The Physics of Bremmstrahlung . 69 5.2 Thermal Bremmstrahlung Emission . 72 5.3 Thermal Bremmstrahlung Absorption . 72 5.4 Relativistic Bremmstrahlung . 73 6 Synchrotron Radiation 77 6.1 Motion in a magnetic field . 77 6.2 Spectrum of Synchrotron radiation . 78 6.2.1 Qualitative Spectrum . 79 6.2.2 Spectral Index for a Power-Law Distribution of Particle Energies . 79 6.2.3 Spectrum and Polarization of Synchrotron emission - Details 80 CONTENTS 5 7 Compton Scattering 87 7.1 The Kinematics of Photon Scattering . 87 7.2 Inverse Compton Scattering . 88 7.2.1 Inverse Compton Power - Single Scattering . 89 7.2.2 Inverse Compton Spectra - Single Scattering . 90 7.3 Repeated Scattering . 92 7.4 Repeated Scattering with Low Optical Depth . 93 8 Atomic Structure 97 8.0.1 A single electron in a central field . 98 8.1 Energies of Electron States . 100 8.2 Perturbative Splittings . 102 8.2.1 Spin-Orbit Coupling . 102 8.2.2 Zeeman Effect and Nuclear Spin . 103 8.3 Thermal Distributions of Atoms . 104 8.3.1 Ionization Equilibrium - the Saha Equation . 104 8.4 A Practical Aside - Order of Magnitude Calculations . 105 9 Radiative Transitions 109 9.1 Perturbation Theory . 109 9.1.1 The Perturbation to the Hamiltonian . 111 9.1.2 Dipole Approximation . 113 9.1.3 Oscillator Strengths . 114 9.2 Selection Rules . 115 9.3 Bound-Free Transitions and Milne Relations . 115 9.4 Line Broadening Mechanisms . 118 10 Molecular Structure 121 10.1 The Born-Oppenheimer Approximation . 121 10.2 Molecular Excitations . 124 11 Fluid Mechanics 127 11.1 Phase-Space Density . 127 11.1.1 Particle Current . 128 11.1.2 Stress Tensor . 129 11.2 Ideal Fluids . 131 11.2.1 Isentropic flows . 132 11.2.2 Hydrostatics . 133 11.2.3 Really Little Sound Waves . 134 11.2.4 Steady Supersonic Flow . 135 11.2.5 Spherical Accretion . 136 12 Unsteady Flows and Applications 139 12.1 Real Sound Waves . 139 12.2 Shock Waves . 141 12.3 A Spherical Shock - The Sedov Solution . 143 6 CONTENTS 12.4 Instabilities . 144 12.4.1 Gravity Waves and Rayleigh-Taylor Instability . 144 12.4.2 Kelvin-Helmholtz or Shearing Instability . 146 12.4.3 Gravitational Instability . 147 Chapter 1 Radiative Transfer 1.1 Introduction We are going to set the stage for a deeper look at astrophysical sources of radia- tion by defining the important concepts of radiative transfer, thermal radiation and radiative diffusion. One can make a large amount of progress by realizing that the distances that radiation typically travels between emission and detection or scattering etc. are much longer than the wavelength of the radiation. In this regime we can assume that light travels in straight lines (called rays). Upon these assumptions the field of radiative transfer is built. 1.2 Flux Let’s start with something familiar and give it a precise definition. The flux is simply the rate that energy passes through an infinitesimal area (imagine a small window). dE = F dAdt (1.1) For example, if you have an isotropic source, the flux is constant across a spher- ical surface centered on the source, so you find that 2 2 E1 = F14πR1andE2 = F24πR2 (1.2) at two radii around the source. Unless there is aborption or scattering between the two radii, E1 = E2 and we obtain the inverse-square law for flux 2 2 F1R1 = F2R2. (1.3) 1.3 Intensity Although the flux is a useful quantity, it cannot encapsulate all of our knowledge about a radiation field. For example, one could shine a faint light directly 7 8 CHAPTER 1. RADIATIVE TRANSFER through a window or a bright light through the same surface at an angle. Both of these siutations are characterized by the same rate of energy flow through the surface, but they are clearly different physical situations. A more generally useful quantity quantifies the rate that energy flows through a surface in a particular direction (imagine that the window now looks into a long pipe so that only photons travelling in a particular direction can pass through. dE = IdAdΩdt (1.4) Although this quantity seems a bit kludgy, it is actually quite familiar. It is the brightness. You look at a light bulb. As you move away from the light bulb, your eye recieves less flux (F decreases) and the apparent size of the light bulb also decreases (dΩ decreases). It turns out that these two quantities both decrease as R−2, so the intensity or brightness is conserved along a ray. This result makes the intensity a terrifically useful quantity. 1.3.1 Relation to the flux From the example at the beginning of this section we can deduce the relation- ship between the flux and the intensity of the light. Radiation that travels perpendicular to a surface delivers more energy to that surface than radiation travelling at an angle. You can always imagine second surface perpendicular to the light ray through which all of the energy that reaches the first surface travels. We know that intensity is the same along the ray so dE = IdA1dΩdt = IdA2dΩdt (1.5) and dA2 = cos θdA1, so the total flux travelling through the surface is given by a moment of the intensity Z F = I cos θdΩ (1.6) If I is constant with respect to angle, there is as much energy travelling from left to right as from right to left, so the net flux vanishes, or more mathematically the mean of cos θ vanishes over the sphere. Something to think about The sun is equally intense in the summer and winter (if you exclude the effects of the atmosphere), then why are winters colder than summers? A closely related quantity is the pressure that a radiation field exerts on a surface. Pressure is the rate that momemtum is delivered to a surface in the direction perpendicular to the surface. The momentum of an individual photon is E/c and the rate that energy is delivered to a surface from photons travelling around a particular direction is simply I cos θdΩ. The component of the momentum that is directed perpendicular to the surface is E cos θ/c, so 1.3. INTENSITY 9 there is a second factor of cos θ yielding the following integral. 1 Z p = I cos2 θdΩ. (1.7) c Something to think about Does the radiation pressure from an isotropic radiation field vanish? 1.3.2 Spectra The quantities that we have defined so far can be examined as a function of the frequency or wavelength of the radiation or the energy of the individual photons, yielding Fν ,FE,Fλ and also for the intensity, e.g dE = Fν dνdAdt. (1.8) The use of Fν is so common that astronomers have a special unit to measure Fν 1 Jansky = 1 Jy = 10−26W m−2Hz−1 = 10−23erg cm−1s−1Hz−1. (1.9) This unit is most commonly used in the radio and infrared, and sometimes in the x-rays. A common combination that people use is dF dF EF = λF = νF = ν = . (1.10) E λ ν dν d ln ν This allows you to convert between Fν and Fλ etc. And it also gives the flux per logarithmic interval in photon energy or frequency.
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