What is the definition of Unit Optical Depth?

In summary, the equation $\chi = n \sigma$ defines the opacity, where Chi is the Opacity, n is the number density of absorbers (constant), and $\sigma$ is the cross section. The optical depth is the number of photon mean-free paths in a given physical step, which can be calculated using the equation $\Delta \tau = \chi \Delta z$. The total optical depth along a straight-line ray is calculated by integrating the equation $\tau = \int_{z_1}^{z_2} \chi \,dz$. The goal is to calculate the distance from an object to Unit Optical Depth at a given wavelength, but the exact definition of Unit Optical Depth is not clear. More information can be
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AlphaCrucis
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Homework Statement
I'm trying to calculate how far away from some object is Unit Optical Depth at some given wavelength; but i'm confused as to what exactly Unit Optical Depth is? What am i Calculating
Relevant Equations
Below
\begin{equation}
\chi = n \sigma
\end{equation}
Where Chi is the Opacity, n is the number density of absorbers (constant), and $\sigma$ is the cross section(given).

We define the optical depth is just the number of photon mean-free paths in a given physical step, i.e. if we consider a physical step:
\begin{equation}
\Delta \tau = \chi \Delta z
\end{equation}

Therefore, in traveling from location z1 to z2 along a straight-line ray, the total optical depth is
\begin{equation}
\tau = \int_{z_1}^{z_2} \chi \,dz
\end{equation}I'm trying to calculate how far away from some object is Unit Optical Depth at some given wavelength; but I'm confused as to what exactly Unit Optical Depth is? What am i calculating? I am given n and the wavelength along with the cross section
 
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Related to What is the definition of Unit Optical Depth?

1. What is unit optical depth?

Unit optical depth is a measure of the amount of light that is absorbed or scattered by a medium. It is a dimensionless quantity that represents the ratio of the incident light intensity to the transmitted light intensity.

2. How is unit optical depth calculated?

Unit optical depth is calculated using the Beer-Lambert law, which states that the amount of light absorbed by a medium is proportional to the concentration of the absorbing material and the distance the light travels through the medium.

3. What is the significance of unit optical depth in scientific research?

Unit optical depth is an important parameter in many fields of science, including atmospheric science, astronomy, and remote sensing. It helps scientists understand the properties of materials and their interactions with light.

4. How does unit optical depth relate to extinction coefficient?

Unit optical depth and extinction coefficient are closely related. The extinction coefficient is a measure of the ability of a material to absorb or scatter light, while unit optical depth is a measure of the amount of light that is absorbed or scattered. They are inversely proportional to each other.

5. Can unit optical depth be negative?

No, unit optical depth cannot be negative. It is a dimensionless quantity and represents the ratio of two positive values (incident light intensity and transmitted light intensity). A negative value would not make physical sense in this context.

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