Calculating the optical depth of an inhomogeous gas

In summary, the question is about calculating the optical depth for an inhomogeneous cloud of radius ##r##. The formula for the optical depth is given by ##\tau = \alpha \frac{\rho}{\rho_c}l## where ##\alpha## is the linear attenuation coefficient, ##\rho## is the density of the cloud, and ##\rho_c## is the density of the condensed phase. This formula is used when the linear attenuation coefficient for the gaseous form is not known. The problem is to calculate the optical depth for a spherical cloud with a finite mass density at ##r = 0##, ##\rho_0##. An attempt at the required integral is given, but there are concerns
  • #1
colorofeternity
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TL;DR Summary
Trying to find the required integral to calculate the optical depth for a cloud with a mass density that follows the inverse square law with a finite density at the center.
My question emerges from my desire to calculate the optical depth, which should be unitless, for an inhomgeneous cloud of radius ##r##. For a homogeneous medium, the optical depth can be defined in terms of the density of a cloud relative to the density of the condensed medium:

$$\tau = \alpha \frac{\rho}{\rho_c}l$$

Where ##\alpha## is the linear attenuation coefficient of light at a given wavelength for the material in question, ##\rho## is the density of the cloud and ##\rho_c## is the mass density of the condensed phase (assuming the cloud is all made of one material, lets say Iron). #l# is the path length of light through the cloud.
This form is used when the linear attenuation coefficient of the gaseous form of the material is unknown, as it is for many metallic gases. What I wish to do is calculate the optical depth for light passing through the spherical cloud of radius, ##r_0## given that the mass density of the cloud follows the inverse-square law with a finite mass density at ##r = 0##, ##\rho_0## (##kg/m^3##).
My attempt at the required integral is as follows:

$$\tau = \int_{0}^{r_0} \frac{\rho_0}{r^2} dr$$

However, I am unsure about this as this expression would give a diverging density at ##r = 0##, which isn't what is happening physically. If I add in an extra ##r^2## term as would be the case for spherical coordinates, then I would end up with a path length of ##r_0##, which seems odd. In addition, it seems that the units don't fully add up as I would expect ##\tau## to be unitless.

How should I approach this problem?
 
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To calculate the optical depth for an inhomogeneous gas, you can use a similar approach as for a homogeneous medium, but instead of using the density of the cloud relative to the density of the condensed medium, you can use the density of the gas at a given point relative to the average gas density. This will give you a unitless optical depth that takes into account the varying density of the gas throughout the cloud.

The integral you have attempted is not correct as it does not take into account the varying density of the gas. Instead, you can use the following expression:

$$\tau = \int_{0}^{r_0} \frac{\rho(r)}{\bar{\rho}} \alpha(r) dr$$

Where ##\rho(r)## is the density of the gas at a given point, ##\bar{\rho}## is the average gas density, and ##\alpha(r)## is the linear attenuation coefficient at that point. This integral will give you the optical depth for light passing through the spherical cloud.

To address the issue of the diverging density at ##r = 0##, you can use a small cutoff radius, ##r_{min}##, and integrate from that radius instead of from ##r = 0##. This will prevent the density from becoming infinite and will still give you a good approximation of the optical depth.

In terms of units, the integral will give you a unitless optical depth, as desired. However, the linear attenuation coefficient, ##\alpha##, will have units of ##m^{-1}##. You can convert this to a unitless coefficient by dividing by the path length, ##l##, giving you a unitless attenuation coefficient, ##\alpha/l##. This can then be used in the integral to calculate the optical depth.

Overall, the key is to take into account the varying density of the gas throughout the cloud and to use a cutoff radius to prevent any diverging values. This approach will give you a more accurate and physically meaningful calculation of the optical depth for an inhomogeneous gas cloud.
 

Related to Calculating the optical depth of an inhomogeous gas

What is optical depth?

Optical depth is a measure of the opacity of a medium, indicating how much light is absorbed or scattered as it travels through the medium. It is a dimensionless quantity that quantifies the attenuation of light intensity due to absorption and scattering.

How do you calculate the optical depth for an inhomogeneous gas?

To calculate the optical depth for an inhomogeneous gas, you need to integrate the extinction coefficient (which includes both absorption and scattering) along the path of the light through the gas. Mathematically, this is expressed as τ = ∫ κ(s) ds, where τ is the optical depth, κ(s) is the extinction coefficient at position s, and ds is a differential element along the path.

What factors affect the optical depth of an inhomogeneous gas?

The optical depth of an inhomogeneous gas is affected by the gas density, the composition of the gas, the wavelength of the light, and the spatial distribution of these properties. Variations in any of these factors along the path of light will influence the overall optical depth.

Why is it important to consider inhomogeneity when calculating optical depth?

Considering inhomogeneity is crucial because the properties of the gas can vary significantly along the path of light. Ignoring these variations can lead to inaccurate calculations of optical depth, which in turn affects the interpretation of observational data, such as the intensity and spectrum of light emerging from the gas.

Can you provide an example of a situation where calculating the optical depth of an inhomogeneous gas is necessary?

One example is in astrophysics, where calculating the optical depth of interstellar gas clouds is necessary to understand the absorption and scattering of starlight. These gas clouds are often inhomogeneous, with varying densities and compositions, making it essential to account for these variations to accurately model the observed light and derive properties of the gas cloud.

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