Radiative transfer equation: Intensity from a disk.

[Tomer]

Hello to everyone who reads, and thank you very much.

1. Problem:
The actual problem is this: A thin disk with thickness 2H and radius R (H<<R), has an absorption co. "alpha" and emision co. j.
Calculate the intensity I("miu", "tau"), where "miu" = cos("theta"), and "theta" is the angle from the disk normal, and "tau" is the optical depth in the vertical direction ("tau" = "alpha" * 2H).

Homework Equations

1. Radiative transfer equation: dI/ds = -"alpha"*I + j (ds = length element along a ray)
2. d("tau") = "alpha" * ds
3. "miu" = cos("theta")

The Attempt at a Solution

My biggest problem is that my understanding of the meaning of intensity is very weak.
I am thinking in the direction of looking at a specific ray coming from a certain "hight" z, compared to the bottom of the disk, coming in angle "theta", and solving the equation above for it, and then summing the effect of all rays, or something.
I could do this and more, but there are some really fundamental things I do not understand and would really appreciate answers:

The transfer equation, as I understand, is describing intensity behaviour for a specific ray, right? But how do I sum up the effects of several rays?
Is it meaningful to ask "what is the intensity at the point P"? What happens if 2 rays with intensity I arrive to a point P? What is the intensity in P then?
We have studied intensity is constant along a ray. How come then, the intensity in any point in space isn't infinite, giving intensity comes from everywhere and remains constant?
I realize these questions might be rather silly and much more basic than the question but I really cannot solve them.
Once I understand these, I might know how to handle the question. I just don't understand if I'm suppose to sum up all the effects of rays going out of the disk in direction "theta" from the normal.


Thank you very very much, I'd really appreciate help here, I'm lost (and I've been reading the book over and over but I just cannot grasp it).

Thanks,
Tomer.

 

[Jhero]

Dear Tomer,

Thank you for sharing your problem with us. It seems like you are struggling with some fundamental concepts in radiative transfer and intensity. I will try my best to address your questions and provide some guidance for solving the problem at hand.

Firstly, let's define some key terms that will help us understand the problem better:

1. Radiative transfer equation: This equation describes the transfer of radiant energy along a ray. It takes into account the absorption and emission of radiant energy by the medium through which the ray is passing.

2. Intensity: In the context of radiative transfer, intensity refers to the amount of radiant energy per unit area per unit solid angle. It is a measure of the strength of the radiant energy passing through a specific point in space.

3. Optical depth: This is a measure of how much a medium can absorb or scatter radiant energy. It is defined as the product of the absorption coefficient and the distance traveled by the ray.

Now, let's move on to your questions:

1. How do I sum up the effects of several rays?
In order to calculate the total intensity at a point, you need to take into account the contribution of all the rays passing through that point. This can be done by integrating the radiative transfer equation over all the rays passing through that point. In simpler terms, you need to add up the intensity of each individual ray to get the total intensity at that point.

2. Is it meaningful to ask "what is the intensity at the point P"?
Yes, it is meaningful to ask for the intensity at a specific point. Intensity is a physical quantity that can be measured at any point in space. However, the intensity at a point depends on the direction and path of the rays passing through that point.

3. What happens if 2 rays with intensity I arrive at point P?
In this case, the total intensity at point P would be 2I, since intensity is additive. However, this does not mean that the intensity at point P is infinite. As the rays travel through the medium, they can lose intensity due to absorption and scattering, resulting in a finite intensity at point P.

4. How come the intensity in any point in space isn't infinite?
As mentioned before, the intensity of a ray can decrease as it travels through a medium due to absorption and scattering. This means that the intensity at a specific point is dependent on the path and direction of the rays passing through that