# Optical depth from radiative transfer equation

• Kayla Martin
In summary, optical depth is a measure of light absorption and scattering in a medium, and it plays a crucial role in radiative transfer, which quantifies the amount of radiation absorbed by a material. The optical depth is calculated by integrating the absorption coefficient over the path length, and it affects the intensity of radiation passing through a medium. Various factors, such as particle concentration and properties, path length, and wavelength, can affect the optical depth. The radiative transfer equation is widely used in fields such as atmospheric science, astrophysics, and remote sensing, as well as in practical applications such as engineering and design.
Kayla Martin
Homework Statement
Hi, I can't figure out how to find the optical depth for the following situation:
A supernova remnant has a brightness of I=1.5x10^{-19} Wm^{-2}Hz^{-1}sr^{-1} at frequencies around 1.6-1.7 GHz, but OH molecules in a homogenous foreground cloud produce an absorption line at 1667MHz. The observed intensity at the center of the line is 3.0x10^{-20} Wm^{-2}Hz^{-1}sr^{-1} and the width is 16kHz (corresponding to a velocity width of about 3km/s).
Assuming T_{ex}=12K throughout the cloud for this transition I need to use the radiative transfer equation to calculate the optical depth at the center of the line, tau_0.
Can someone please help me figure this out? I know we have been given all the equations for this in our lecture notes, but I am stumped at how to put it all together?
Relevant Equations
$$\frac{dI}{dS} = - \alpha I + j$$ where $$\alpha$$ is absorption coefficient...
$$\frac{dI}{\tau} = - I + S$$ where S is the source function $$S = \frac{j}{\alpha}$$ and $$\tau$$ is the optical depth.

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The only solution (for part a) I can think of is if we use the radiative transfer equation without S... i.e. $$\frac{I}{\tau}= I_0 e^{-\tau}$$ and then take the natural log of each side to get $$\tau= -\ln(\frac{I}{I_0})$$ but I don't know if I can just get rid of S like that?

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## 1. What is the radiative transfer equation?

The radiative transfer equation is a mathematical equation that describes the transfer of electromagnetic radiation through a medium. It takes into account factors such as absorption, emission, and scattering of radiation by particles in the medium.

## 2. How is optical depth defined?

Optical depth is a measure of the amount of absorption and scattering of radiation by a medium. It is defined as the natural logarithm of the ratio of the incident radiation to the transmitted radiation through the medium.

## 3. What is the significance of optical depth in radiative transfer?

Optical depth is an important parameter in radiative transfer because it determines the amount of radiation that is absorbed or scattered by a medium. It is also used to calculate the radiative flux and the energy balance in a system.

## 4. How is optical depth calculated from the radiative transfer equation?

To calculate the optical depth from the radiative transfer equation, one needs to know the extinction coefficient, which is a measure of the ability of a medium to absorb and scatter radiation. The optical depth is then calculated by integrating the extinction coefficient over the path length of the radiation through the medium.

## 5. What are some applications of the radiative transfer equation and optical depth?

The radiative transfer equation and optical depth have various applications in atmospheric and environmental sciences, such as in climate modeling, remote sensing, and atmospheric chemistry studies. They are also used in astrophysics to understand the properties of interstellar medium and to analyze the spectra of stars and galaxies.

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