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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?

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

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?