Stellar Astrophysics Final Study Guide

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Key Equations & Principles (Understand Derivation & Application) Refer to MG: Chapters 1-10, 13, 14, 16, 17 and LB: Chapters 1-6 for detailed derivations and explanations. 1. Stellar Structure Equations Mass Conservation: $\frac{dm}{dr} = 4\pi r^2 \rho$ Hydrostatic Equilibrium: $\frac{dP}{dr} = -\frac{Gm\rho}{r^2}$ Energy Generation: $\frac{dL}{dr} = 4\pi r^2 \rho \epsilon$ Energy Transport (Radiative): $\frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho}{T^3} \frac{L}{4\pi r^2}$ Energy Transport (Convective, adiabatic): $\frac{dT}{dr} = (1 - \frac{1}{\gamma}) \frac{T}{P} \frac{dP}{dr}$ 2. Virial Theorem For a bound system: $2K + U = 0$ (where $K$ is kinetic energy, $U$ is potential energy) Implications for stellar collapse and stability. 3. Stellar Timescales Free-fall: $t_{ff} \approx (G\rho)^{-1/2}$ Thermal (Kelvin-Helmholtz): $t_{KH} \approx \frac{GM^2}{RL}$ Nuclear: $t_{nuc} \approx \frac{E_{nuc}}{L}$ 4. Radiative Transfer Specific Intensity: $I_\nu$ Source Function: $S_\nu = \frac{j_\nu}{\kappa_\nu \rho}$ Equation of Radiative Transfer: $\frac{dI_\nu}{ds} = -\kappa_\nu \rho I_\nu + j_\nu = -\kappa_\nu \rho (I_\nu - S_\nu)$ Optical Depth: $d\tau_\nu = -\kappa_\nu \rho ds$ Formal Solution (Plane-Parallel): $I_\nu(\tau_\nu=0) = \int_0^\infty S_\nu e^{-\tau_\nu} d\tau_\nu$ 5. Blackbody Radiation Planck's Law: $B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT}-1}$ Stefan-Boltzmann Law: $L = 4\pi R^2 \sigma T_{eff}^4$ Wien's Displacement Law: $\lambda_{max} T = b$ 6. Atomic Physics for Spectral Lines Einstein Coefficients: $A_{ul}, B_{lu}, B_{ul}$ (for spontaneous emission, stimulated absorption, stimulated emission) Boltzmann Equation: $\frac{N_u}{N_l} = \frac{g_u}{g_l} e^{-\Delta E/kT}$ Saha Equation: For ionization balance.