Radiative Transfer Parallel Plane Atmosphere Help

1. Apr 9, 2013

darkeinstein

I've been working on this problem for about a week (mostly trying to understand it), I'm making little progress and it's due tomorrow. Any help or hints would be greatly appreciated.

It's a long paragraph of a problem, so I'll try to summarize as best I can...

Main Question: Derive an expression for the physical depth to which we can see into the atmosphere, as a function of wavelength.

Plane parallel atmosphere, z-axis has z=0 at surface, increases going into star. θ=0 is the observing angle from z, (same as z).

Constant density ρ0, Temperature: T(z) = T0(z/H0)

Opacity: κλ = κ0 + κ1*e^[(-(λ-λ0)^2)/2σ^2]

κ0 = continuum opacity & κ1 = opacity at λ0

κ1 has a Gaussian distribution with width σ around λ0

Assuming we can see to optical depth of τ ~ 1, derive an expression for the physical depth to which we can see into the atmosphere, as a function of wavelength.

Show when your looking at a wavelength far from λ0, you can see a factor of 1 + (k1/k0) deeper into the atmosphere compared to λ0.

Assume Temp is blackbody.

Assume wavelength range is far from peak, so can use Rayleigh-Jeans approximation, & assume you can replace λ with λ0 here. (ignore variation of intensity with wavelength for bb radiation, focus on wavelength dependence of opacity.) - makes background cont. flat.

Plot I(λ)/I(λ0) as a function of ζ = (λ-λ0)/σ, assuming k1/k0 = 2. Is this an absorption or emission line?

END OF QUESTION....................................WHEW!

Relevant equation(s):

Iλ = Bλ(T) + cosθ(dBλ/dτλ)

Vertical Optical Depth: τλ,v(z) = ∫κλρdz

His one hint on the question was: κρs = τ = 1, find distance (s), stick it in function to get temp, stick in BB to get Intensity. (s should = z since s is the distance at an angle, but θ=0)

I don't know much of any of this, but I'd mostly appreciate help with deriving the physical depth equation.

Here's what I can figure so far...

Physical depth as function of λ, should be: z(λ)

z(λ) = (T*H(not)/T(not)) ρ(1 + (κ1/κ(not))) since κ is a function of wavelength

since T(z) = T(not)(z/(H(not))).

I'm not really sure what to do, I'm looking at 20 or so different equation, and I'm not really sure how to get depth as a function of wavelength or where to start.

Any help or hints would be greatly appreciated, thanks.