Astro 300B: Jan. 21, 2011 Equation 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

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

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

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 in out + p ┴ - p ┴

Since the radiation field is isotropic, J ν = I ν Integrate over 2π steradians only Not 2π

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

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

F ν = π B ( 1 – cos 2 θ c ) = π B sin 2 θ c Or…. Note: at r = R

Equation of Radiative Transfer When photons pass through material, I ν changes due to (a) absorption (b) emission (c) scattering ds IνIν I ν + dI ν dI ν = dI ν + - dI ν - - dI ν sc I ν added by emission I ν subtracted by absorption I ν subtracted by scattering

EMISSION: dI ν + DEFINE j ν = volume emission coefficient j ν = 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

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

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

Microscopic Picture N absorbers / cm 3 Each absorber has cross-section for absorption has units cm 2 ; is a function of frequency ASSUME: (1)Randomly distributed, independent absorbers (2) No shadowing:

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

Can also define the mass absorption coefficient Where ρ = mass density, ( g cm -3 ) has units cm 2 g -1 Sometimes is denoted

So… the Equation of Radiative Transfer is OR emission absorption Amount of I ν removed by absorption is proportional to I ν Amount of I ν added by emission is independent of I ν TASK: find α ν and j ν for appropriate physical processes

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

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

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

General Solution L1L1 L2L2 Eqn. 1 Multiply Eqn. 1 by And rearrange Eqn. 2

Now integrate Eqn. 2 from L 1 to L 2 LHS= So…

at L2 is equal to the incident specific intensity, decreased by a factor of plus the integral of j ν along the line of sight, decreased by a factor of = the integral of α ν from l to L 2