__Question__: Can you clarify the meaning of 'local thermodynamic equilibrium' in relation to black body radiation for a star? (P.S. I

*am*a math whiz!) - Margith, Salt Lake, UT

__Answer__: It will be of use here to refer to the diagram below, which depicts the transfer of radiation for a star treated as having a "plane parallel" atmosphere, i.e. the stellar layers are essentially parallel;

Of importance here is The Planck function describes the distribution of radiation for a

**black bod**

**y**, and can be expressed:

^{2})/ l

^{5}} [1/ exp (hc/lkT) - 1)]

**l**:

dI(l)/ds = -k(l) I(l) + k(l) S(l) = k(l) [S(l) – I(l)] - 0 or I(l) = S(l)

**I(l) equals the Planck function:**

*a black body,***if**LTE holds, the photons always emerge at all wavelengths. In the above treatment, note that the absorption coefficient was always written as: k(l) to emphasize its wavelength (l) dependence.

p( F

_{o}) = 2 p (I(cos (q)) = p [a(l) + 2(b(l)/3 ]

and: F

_{lo}= S(l) t(l) = 2/3

This states that the flux coming

*out of the stellar surface*is equal to the source function at the*optical depth*t*= 2/3*. This is the very important ‘*Eddington-Barbier’ relation*that facilitates an understanding of how stellar spectra are formed.
Once one then assumes LTE, one can further assume k(l) is independent of l (gray atmosphere) so that:

k(l) = k; t (l) = t and F

k(l) = k; t (l) = t and F

_{lo}= B_{l}(T(t = 2/3) )
Thus, the energy distribution of F

_{l}is that of a black body corresponding to the temperature at an optical depth t = 2/3. From this, along with some simple substitutions and integrations a wide array of problems can be done. Among the more interesting applications - since you are a math whiz - is to find the "equivalent width", e.g. of a spectral line:
W =

**ò**^{¥}**(I**_{0 }_{c}– I_{u }/ I_{c})du =**ò**^{¥}**(F**_{0 }_{c}– F_{l }/ F_{c}) dl
The left side defines W in terms of the intensity from the continuous spectrum outside the spectral line where the quantity (I

_{c}– I_{u }/ I_{c}) is referred to as the*“depth*of the line”, the analogous quantity to (F_{c}– F_{l }/ F_{c}) on the right side where we have radiant flux units. Technically, the integral should be taken only from one side of the line to the other but the limits can be as shown provided I_{c}(or F_{c}) is kept constant in the neighborhood of the line.
## No comments:

Post a Comment