Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Radiative Transfer Parallel Plane Atmosphere Help

  1. Apr 9, 2013 #1
    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.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted