Astro 300B: Jan. 21, 2011 Equation of Radiative Transfer

1. Astro 300B: Jan. 21, 2011Equation of Radiative Transfer Sign Attendance Sheet Pick up HW #2, due Friday Turn in HW #1 Chromey,Gaches,Patel: Do Doodle poll First Talks: Donnerstein, Burleigh, Sukhbold Fri., Feb 4

2. Radiation Energy Density uν specific energy density uν = energy per volume, per frequency range Consider a cylinder with length ds = c dtc = speed of light dA uν(Ω) = specific energy density per solid angle Then dE = uν(Ω) dV dΩ dν Hz ds energy steradian volume But dV = dA c dtfor the cylinder, so dE = uν(Ω) dA c dt dΩ dν Recall that dE = Iν dA dΩ dt dν so….

3. Integrate uν(Ω) over all solid angle, to get the energy density uν Recall so ergs cm-3 Hz-1

4. Radiation Pressure of an isotropic radiation field inside an enclosure What is the pressure exerted by each photon when it reflects off the wall? Each photon transfers 2x its normal component of momentum photon + p┴ in out - p┴

5. Since the radiation field is isotropic, Jν = Iν Integrate over 2π steradians only Not 2π

6. But, recall that the energy density So…. Radiation pressure of an isotropic radiation field = 1/3 of its energy density

7. Example: Flux from a uniformly bright sphere (e.g. HII region) θ P At point P, Iν from the sphere is a constant (= B) if the ray intersects the sphere, and Iν = 0 otherwise. R θc r So we integrate dφ from 2π to 0 And…looking towards the sphere from point P, in the plane of the paper dφ

8. Fν = π B ( 1 – cos2θc ) = π B sin2θc Or…. Note: at r = R

9. Equation of Radiative Transfer When photons pass through material, Iνchanges due to (a) absorption (b) emission (c) scattering ds Iν + dIν Iν Iν subtracted by scattering dIν = dIν+- dIν- - dIνsc Iν subtracted by absorption Iνadded by emission

10. EMISSION: dIν+ DEFINE jν = volume emission coefficient jν = energy emitted in direction Ĩ per volume dV per time dt per frequency interval dν per solid angle dΩ Units: ergs cm-3 sec-1 Hz-1 steradians-1

11. Sometimes people write emissivity εν = energy emitted per mass per frequency per time integrated over all solid angle So you can write: or Fraction of energy radiated into solid angle dΩ Mass density

12. ABSORPTION Experimental fact: Define ABSORPTION COEFFICIENT = such that has units of cm-1

13. Microscopic Picture N absorbers / cm3 Each absorber has cross-section for absorption has units cm2; is a function of frequency • ASSUME: • Randomly distributed, independent absorbers • (2) No shadowing:

14. Then Total # absorbers in the volume = Total area presented by the absorbers = So, the energy absorbed when light passes through the volume is In other words, is often derivable from first principles

15. Can also define the mass absorption coefficient Where ρ = mass density, ( g cm-3 ) has units cm2 g-1 Sometimes is denoted

16. So… the Equation of Radiative Transfer is OR absorption emission Amount of Iνremoved by absorption is proportional to Iν Amount of Iνadded by emission is independent of Iν TASK: find αν and jν for appropriate physical processes

17. Solutions to the Equation of Radiative Transfer (1) Pure Emission (2) Pure absorption (3) Emission + Absorption

18. Pure Emission Only Absorption coefficient = 0 So, Increase in brightness = The emission coefficient integrated along the line of sight. Incident specific intensity

19. (2) Pure Absorption Only Emission coefficient = 0 incident Factor by which Iν decreases = exp of the absorption coefficient integrated along the line of sight

20. General Solution L2 L1 Eqn. 1 And rearrange Multiply Eqn. 1 by Eqn. 2

21. Now integrate Eqn. 2 from L1 to L2 LHS= So…

22. at L2 is equal to the incident specific intensity, decreased by a factor of plus the integral of jνalong the line of sight, decreased by a factor of = the integral of αν from l to L2