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Published byAmelia Roberts Modified over 6 years ago
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The Transfer Equation The basic equation of transfer for radiation passing through gas: the change in specific intensity In is equal to: dIl = intensity emitted – intensity absorbed dIl = jlrdl – klrIl dl -dIl /dtl = Il - jl/kl = Il - Sl This is the basic equation which must be solved to compute the spectrum emerging from or passing through a gas.
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Thermodynamic Equilibrium
Every process of absorption is balanced by a process of emission; no energy is added or subtracted from the radiation Then the total flux is constant with depth flux is the energy passing through a unit surface area integrated over all directions mean intensity is the directional average of the specific intensity When we assume LTE, we are assuming that Sl=Bl
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Simplifying Assumptions
Plane parallel atmospheres (the depth of a star’s atmosphere is thin compared to its radius, and the MFP of a photon is short compared to the depth of the atmosphere Opacity is independent of wavelength (a gray atmosphere)
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Black Bodies - Observations
spectrum continuous, isotropic, unpolarized continuum intensity depends on frequency and temperature observed relation: From this can be derived Wien’s law and the Stefan-Boltzman law Also Rayleigh-Jeans Approx. and Wien Approx.
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Black Bodies Rayleigh-Jeans approximation Wien approximation
Wien’s Law – Peak intensity Stefan-Boltzman Law – Luminosity Planck’s Law – Energy Distribution Rayleigh-Jeans approximation Wien approximation
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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.
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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?
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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
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Steps to the Planck Function
Energy level populations given by the Boltzman equation: Include spontaneous and stimulated emission Solve for I, substitute Nu/Nl Note that
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Planck’s Law Rayleigh-Jeans Approximation (at long wavelength, hn/kT is small, ex=x+1) Wien Approximation – (at short wavelength, hn/kT is large)
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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.
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Using Planck’s Law Computational form:
For cgs units with wavelength in Angstroms
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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?
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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 proto-stellar object with a luminosity 50 times greater than the Sun and a diameter of 3” at a distance of 200 pc.
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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?
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