Optical depth (astrophysics)

Optical depth in astrophysics refers to a specific level of transparency. Optical depth and actual depth, τ {displaystyle au } and z {displaystyle z} respectively, can vary widely depending on the absorptivity of the astrophysical environment. Indeed, τ {displaystyle au } 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 in astrophysics refers to a specific level of transparency. Optical depth and actual depth, τ {displaystyle au } and z {displaystyle z} respectively, can vary widely depending on the absorptivity of the astrophysical environment. Indeed, τ {displaystyle au } 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. The assumption here is that either the extinction coefficient α {displaystyle alpha } or the column number density N {displaystyle N} 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. From the definition, it is also clear that large optical depths correspond to higher rate of obscuration. Optical depth can therefore be thought of as the opacity of a medium. The extinction coefficient α {displaystyle alpha } can be calculated using the transfer equation. In most astrophysical problems, this is exceptionally difficult to solve since solving the corresponding equations requires the incident radiation as well as the radiation leaving the star. These values are usually theoretical. In some cases the Beer-Lambert Law can be useful in finding α {displaystyle alpha } . where κ {displaystyle kappa } is the refractive index, and λ 0 {displaystyle lambda _{0}} is the wavelength of the incident light before being absorbed or scattered. It is important to note that the Beer-Lambert Law is only appropriate when the absorption occurs at a specific wavelength, λ 0 {displaystyle lambda _{0}} . For a gray atmosphere, for instance, it is most appropriate to use the Eddington Approximation. Therefore, τ {displaystyle au } is simply a constant that depends on the physical distance from the outside of a star. To find τ {displaystyle au } at a particular depth z ′ {displaystyle z'} , the above equation may be used with α {displaystyle alpha } and integration from z = 0 {displaystyle z=0} to z = z ′ {displaystyle z=z'} . Since it is difficult to define where the photosphere of a star ends and the chromosphere begins, astrophysicists usually rely on the Eddington Approximation to derive the formal definition of τ = 2 / 3 {displaystyle au =2/3} Devised by Sir Arthur Eddington the approximation takes into account the fact that H − {displaystyle 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,