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Optical depth in astrophysics refers to a specific level of transparency. Optical depth and actual depth, $\tau$ and $z$ respectively, can vary wildly depending on the absorptivity of the stellar interior. Because of this $\tau$ is able to show the relationship between these two quantities and can lead to a greater understanding of the structure inside a star.

Optical depth is a measure of the extinction coefficient or absorptivity up to a specific 'depth' of a star's makeup.

$\tau \equiv \int _{0}^{z}(\alpha )dz=\sigma N$

This equation assumes that the extinction coefficient $\alpha$ is known, or that N, the column density, is known. These can generally be calculated from other equations if a fair amount of information is known about the chemical makeup of the star.

$\alpha$ can be calculated using the transfer equation. In most astrophysics problems this is exceptionally difficult to solve, since the equations assume one knows the incident radiation as well as the radiation leaving the star and these values are usually theoretical.

In some cases the Beer-Lambert Law can be useful in finding $\alpha$.

$\alpha =e^{\frac {4\pi \kappa }{\lambda _{0}}}$

where $\kappa$ is the refractive index, and $\lambda _{0}$ is the wavelength of the incident light before being absorbed or scattered. Note that the Beer-Lambert Law is only appropriate when the absorption occurs at a specific wavelength, $\lambda _{0}$, for a gray atmosphere it is most appropriate to use the Eddington Approximation.

Therefore it is straightforward to see that $\tau$ is simply a constant that depends on the physical distance from the outside of a star. To find $\tau$ at a particular depth z, one simply uses the above equation with $\alpha$ and integrates from $z=0$ to $z=z$.

The Eddington Approximation and the Depth of the Photosphere

Because it is difficult to define where the photosphere of a star ends and the chromosphere begins astrophysicists rely on the Eddington Approximation to derive the formal definition of $\tau ={\frac {2}{3}}$

Devised by Sir Arthur Eddington the approximation takes into account the fact that $H^{-}$ produces a "gray" absorption in the atmosphere of a star, that is, it is independent of any specific wavelength and absorbs along the entire electromagnetic spectrum. In that case,

$T^{4}={\frac {3}{4}}T_{e}^{4}\left(\tau +{\frac {2}{3}}\right)$

Where $T_{e}$ is the effective temperature at that depth and $\tau$ is the optical depth.

This illustrates not only that the observable temperature and actual temperature at a certain physical depth of a star vary, but that the optical depth plays a crucial role in understanding the stellar structure. It also serves to demonstrate that the depth of the photosphere of a star is highly dependent upon the absorptivity of its environment. The photosphere extends down to a point where $\tau$ is about 2/3, which corresponds to a state where a photon would experience, in general, less than 1 scattering before leaving the star.

One should also note that the above equation can be rewritten in terms of $\alpha$ in the following way:

$T^{4}={\frac {3}{4}}T_{e}^{4}\left(\int _{0}^{z}(\alpha )dz+{\frac {2}{3}}\right)$

Which is useful if $\tau$ is not known but $\alpha$ is.

References

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