Optical depth (astrophysics)
Optical depth in astrophysics refers to a specific level of transparency. Optical depth and actual depth, and respectively, can vary wildly depending on the absorptivity of the stellar interior. Because of this is able to show the relationship between these two quantities and can lead to a greater understanding of the structure inside a star.
This equation assumes that the extinction coefficient 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.
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 .
where is the refractive index, and 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, , for a gray atmosphere it is most appropriate to use the Eddington Approximation.
Therefore it is straightforward to see that is simply a constant that depends on the physical distance from the outside of a star. To find at a particular depth z, one simply uses the above equation with and integrates from to .
The Eddington Approximation and the Depth of the Photosphere
Devised by Sir Arthur Eddington the approximation takes into account the fact that 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,
- Where is the effective temperature at that depth and 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 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 in the following way:
Which is useful if is not known but is.