Theory of the Fireball

T-R -10
(3.16)
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The numerical calculations of Brode are in good agreement with this at
the relevant times, from about 0.05 to 0.5 second.
It will be noted that (3.16) was obtained without integration of
the hydrodynamic equations; it follows simply from the equation of state.
The weakest assumptions are (1) the relation between E and T, (3.13), and
(2) the neglect of the constant in (3.9) . But in any case, T will be a
very hign power of R.
C. Mean Free Path and Radiating Temperature
The emission of radiation from a sphere of variable temperature is
governed by the absorption coefficient. For visible light, the absorption
coefficient increases rapidly with temperature. For any given wave
length, tine emission w ill then come from a layer which is one optical
mean free path inside the hot material.*
*Actually the maximum emission comes from deeper inside the fireball.
To see this we compute the ernigsion normal to the surface, J =
2hv3 e -hv@
jdr@(R)e -D(R)dE? where e = - is the emissivity, and the remain-
C2.e
ing notation is as in the text (below). The integrand has a maximum at
R"/$* = a + cuhv/fl*. The optical depth at the maximum is D* = D(R*) =
ncu + @nv'kir* which is about 2, rather than 1 in the blue. By steepest
a-1
descents and with occasional use of 1/m = 0 it turns out that J =
e 4- B( v,T*), greater by about a factor 2 than the J used in
the text.
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