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.

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

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)

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.

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

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: Include spontaneous and stimulated emission Solve for I, substitute Nu/Nl Note that

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)

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: 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 proto-stellar 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?