Theory of the Fireball

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dF: = - 4KaT
at
where a is the Stefan-Boltzmann constat and K tne opacity in cm /gm,
(7.5)
i.e., the absorption coefficient averaged over a PlaJnck spectrum (dis-
regarding spectral regions which are still black) . Since K increases
rapidly with temperature, the hottest region near the center will cool
fastest, so that tne temperature tends to become more uniform.
For the same reason, the radiative coolhg will effectively stop at
a temperature of about 5000°. At t‘nis temperature and p = O.Olpo, it
takes about 5 seconds to cool the fireball by LO$. At 4000°, this takes
about 200 seconds. Gilmore2’ ‘has calculated curves of cooling times for
transparent bodies at various densities as a function of tine final temper-
ature.
Depending on the ambient density, tne pressure may or may not have
decreased to ambient pressure when the temperature has decreased to ‘OOOo.
Even if it has not, the subsequent adiabatic expansion t rill not lower the
temperature much further. (Our theory is not applicable to very lar~
ambient densities because there the isolation of isotnermal sphere from
t’ne outside never takes place. Probably the limit of applicability is
-2
about pl/po = 10 . Therefore, and because of the small value of y - 1,
expansion cannot be large.) Therefore, after both radiation and hydro-
dynamics have effectively stopped, the fireball is left at a temperature
not much belov 5000°.
Any furt‘ner cooling can only be achieved by the rise of the fireball
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2
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