Optical depth (astrophysics)

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This article is about optical depth in astrophysics. For optical depth in general, see optical depth.

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 [1]

This equation assumes that the extinction coefficient \alpha is known, or that N, the column number 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.[2] 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[edit]

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.