Relating Optical Depth and Extinction (τ(λ) and A(λ))

In summary, the relationship between extinction A(λ) and optical depth τ(λ) can be expressed as A(λ)=1.086*τ(λ). This can be shown using the definition of a difference in apparent magnitudes and the relationship between intensities and magnitudes. By rearranging the equation for m(λ), it can be seen that A(λ) is equal to the difference between the apparent and absolute magnitudes. By relating this to the intensity of light, it can be shown that A(λ) is equal to 1.086 times τ(λ). This can be used to solve for the relationship between τ(λ) and A(λ) in a given scenario.
  • #1
Mike89
4
0
Hey guys sorry my first post is a help request and not an introduction or anything but I'd really appreciate a helping hand here

Homework Statement



Show that the extinction A(λ) and optical depth τ(λ) are related by the linear relationship A(λ)=1.086*τ(λ)

Homework Equations



m(λ)=M(λ) + 5Log(d)-5 + A(λ)
which I'd rearrage to
m(λ)-M(λ)-5Log(d)+5=A(λ)

Here in lies the problem, I have all the equations I can find in my notes and the slides but nothing that relates τ(λ) to anything other than:

exp(-τ) = (I/I0) (sorry wasn't sure how to subscript the 0 in I0)

and then nothing to relate intensities to magnitudes.

The Attempt at a Solution



I thought maybe I could use 2 stars where I new the apparent and absolute magnitudes ( the sun and vega, etc) and the distance obviously and then calculate how τ(λ) and A(λ) are related and show the 1.086 multiplier in both but I can't work out how to relate A(λ) to anything involving intensity.


I hope you guys can help me out here since I'm stumped :) thanks for anything you can
 
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  • #2
Start with the definition of a difference in apparent magnitudes. . .

[tex] m_1 - m_2 = 2.5log(I_1/I_2) [/tex]

extinction, A is defined as [tex] A = m^{'} - m [/tex]

Now, how are I_1, I_2, and tau related. . .
 
  • #3
AstroRoyale said:
[tex] m_1 - m_2 = 2.5log(I_1/I_2) [/tex]

Souldn't it be [tex] m_1 - m_2 = 2.5log(I_2/I_1) [/tex] ?
 

Related to Relating Optical Depth and Extinction (τ(λ) and A(λ))

1. What is optical depth?

Optical depth, denoted by τ(λ), is a measure of the level of absorption and scattering of light as it passes through a medium, such as a gas or a cloud. It is a dimensionless quantity that ranges from 0 (no absorption or scattering) to infinity (complete absorption or scattering).

2. How is optical depth related to extinction?

Optical depth and extinction (A(λ)) are closely related, as both measure the attenuation of light as it passes through a medium. However, while optical depth is a measure of the total amount of attenuation, extinction specifically refers to the loss of light due to absorption by the medium.

3. What factors affect the optical depth and extinction of a medium?

The optical depth and extinction of a medium are affected by several factors, including the density and composition of the medium, the wavelength of the light, and the geometry of the medium (i.e. thickness and size of particles).

4. How are optical depth and extinction measured?

Optical depth and extinction are typically measured using specialized instruments, such as spectrophotometers or lidar systems, which can measure the intensity of light before and after it passes through a medium. These measurements can then be used to calculate the optical depth and extinction at different wavelengths.

5. What are the applications of studying the relationship between optical depth and extinction?

The relationship between optical depth and extinction is important in various fields, including atmospheric science, remote sensing, and astronomy. It can provide insights into the properties of different media, such as the composition and density of gases in Earth's atmosphere, or the size and distribution of particles in space. Understanding this relationship can also help in the development of technologies for measuring and monitoring these properties.

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