Transcript Note
A short review • The basic equation of transfer for radiation passing through gas: the change in specific intensity In is equal to: -dIl /dtl = Il - jl/kl = Il - Sl • Assumptions – LTE… (all physical processes in balance) Sl = Bl – Flux constant with depth – Plane parallel atmosphere Black Body Radiation • walls heated • emit and reabsorb radiation • interior of chamber in thermodynamic equilibrium • leakage in or out of the whole is small • Stellar photospheres approximate blackbodies • Most photons are reabsorbed near where they are emitted • Higher in the atmosphere, a star departs from a black body See: http://homepages.wmich.edu/~korista/phys325.html See: http://www.jb.man.ac.uk/distance/life/sample/stars/index.html See: http://astro1.panet.utoledo.edu/~ndm/4810.html Black Bodies - Observations • spectrum continuous, isotropic, unpolarized • continuum intensity depends on frequency and temperature • observed relation: 3 n In n f T • From this observational result can be derived Wien’s law (peak intensity) and the Stefan-Boltzman law (luminosity) • Also Rayleigh-Jeans approx. and Wien approx. of flux above and below BB peak Wien’s Law – Peak Intensity Il is max at lmax = 0.29/T (l in cm) or l’max = 0.51/T (where l’max is the wavelength at which In is max) Thought Problem: Calculate the wavelengths at which In and Il are maximum in the Sun. Think about why these are different. Luminosity – Stefan Boltzman Law • F = sT4 or L = 4p R2 sT4 • Class Problem: What is the approximate absolute magnitude of a DA white dwarf with an effective temperature of 12,000, remembering that its radius is about the same as that of the Earth? – what is the simplest approach? Deriving the Planck Function • Several methods (2 level atom, atomic oscillators, thermodynamics) • Use 2-level atom: Einstein Coefficients – Spontaneous emission proportional to Nn x Einstein probability coefficient jnr = NuAulhn – Induced (stimulated) emission proportional to intensity knrIn = NlBluInhn – NuBulInhn Steps to the Planck Function • Energy level populations given by the Boltzman equation: N u g n hn / kT e Nl gl • Include spontaneous and stimulated emission Nu Aul Nu Bul In Nl Blu In • Solve for I, substitute Nu/Nl • Note that g Bul Blu l gu 2hn 3 Aul 2 Bul c 2hn 3 1 Bn (T ) 2 hn / kT c e 1 2 2hc 1 Bl (T ) 5 hc / lkT l e 1 Planck’s Law • Rayleigh-Jeans Approximation (at long wavelength, hn/kT is small, ex=x+1) In 2kT n2 c 2 2ktl2 • Wien Approximation – (at short wavelength, hn/kT is large) 2hn 3 hn / kT In 2 e c Class Problem • The flux of M3’s IV-101 at the K-band is approximately 4.53 x 105 photons s–1 m–2 mm-1. What would you expect the flux to be at 18 mm? The star has a temperature of 4250K. Using Planck’s Law Computational form: Bl (T ) 1.19 x10 27 l5 1.44 x108 / lT e 1 For cgs units with wavelength in Angstroms Class Problems • You are studying a binary star comprised of an B8V star at Teff = 12,000 K and a K2III giant at Teff = 4500 K. The two stars are of nearly equal V magnitude. What is the ratio of their fluxes at 2 microns? • In an eclipsing binary system, comprised of a B5V star at Teff = 16,000K and an F0III star at Teff = 7000K, the two stars are known to have nearly equal diameters. How deep will the primary and secondary eclipses be at 1.6 microns? Class Problems • Calculate the radius of an M dwarf having a luminosity L=10-2LSun and an effective temperature Teff=3,200 K. What is the approximate density of this M dwarf? • Calculate the effective temperature of a protostellar object with a luminosity 50 times greater than the Sun and a diameter of 3” at a distance of 200 pc. Class Problems • You want to detect the faint star of an unresolved binary system comprising a B5V star and an M0V companion. What wavelength regime would you choose to try to detect the M0V star? What is the ratio of the flux from the B star to the flux from the M star at that wavelength? • You want to detect the faint star of an an unresolved binary system comprising a K0III giant and a DA white dwarf with a temperature of 12,000 K (and MV=10.7). What wavelength regime would you choose to try to detect the white dwarf? What is the ratio of the flux from the white dwarf to the flux from the K giant at that wavelength?