Radiative Transfer Parallel Plane Atmosphere Help

[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: κ_λ = 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):

I_λ = B_λ(T) + cosθ(dB_λ/dτ_λ)

Vertical Optical Depth: τ_λ,v(z) = ∫κ_λρdz


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.

 

[AndyW]

I understand that you are struggling with understanding and solving this problem. I am happy to offer some guidance and help you progress towards a solution.

Firstly, it's important to break down the problem into smaller, more manageable parts. Let's start by looking at the main question: deriving an expression for the physical depth as a function of wavelength.

To do this, we need to consider the relevant equations provided in the problem. One key equation is the vertical optical depth, τλ,v(z), which represents the amount of light absorbed or scattered as it travels through the atmosphere. We can express this as:

τλ,v(z) = ∫κλρdz

This equation tells us that the vertical optical depth is equal to the integral of the opacity (κλ) multiplied by the density (ρ) over the distance (z) in the atmosphere. We also know that the optical depth is related to the intensity of the light, which can be expressed as:

Iλ = Bλ(T) + cosθ(dBλ/dτλ)

This equation tells us that the intensity of the light at a given wavelength (λ) is equal to the blackbody emission (Bλ) plus the change in intensity due to the absorption or scattering (cosθ(dBλ/dτλ)).

Now, let's look at the hint provided: κρs = τ = 1. This means that at a certain distance (s), the optical depth is equal to 1. We can use this information to find the distance (s) and then plug it into the equations to solve for the temperature (T) and intensity (Iλ).

Next, we need to consider the assumptions made in the problem. We are assuming a plane-parallel atmosphere and a blackbody temperature distribution. We are also using the Rayleigh-Jeans approximation, which is valid for wavelengths far from the peak of the blackbody spectrum. This means we can replace λ with λ0, the central wavelength in the opacity distribution.

Now, we can use the equation for the vertical optical depth to find the distance (s) at which τ = 1. This will give us:

s = 1/κρ

Since we know that s = z at an angle of θ = 0, we can substitute z for s in the equation above. This gives us:

z = 1/κρ

Next, we can use the temperature