Theory of the Fireball

shock, Eq. (3.15) . Next comes t'ne cooling wave I11 in whicn t e temper-
ature falls more steeply, according to (3.8) . (At late times, region I1
is wiped out and I11 follows immediately upon I . ) Region IV includes
the material which has gone through tne cooling wave, and now cools adi-
abatically; nence the temperature falls slowly with r (Sec . 'jd) D is
the material point from whicn the cooling wave started originally; region
V, outside that point, is also expanding adiabatically, but from snock
conditions; thus it is the continuation of region 11. Finally region VI
is the air not yet shocked. As time goes on, the cooling wave I11 moves
inward, wiping out region I1 and then eating into region I. Region IV
accordingly grows toward the inside, but its outer end D stays fixed.
Region V expands into VI by shock.
We note once more that u is the velocity in Lagrange coordinates,
2
and in gm/cm sec. The problem is xnade somewhat more complicated by the
three dimensions and the adiabatic expansion, cf. Sec. 6, but tne prin-
cipal features remain the same
d. Adiabatic Expansion after Cooling. Radiating Temperature
When a given material element has gone through the cooling wave, it
is left at the radiating temperature T1. Thereaf'ter, it w ill expand
adiabatically. Equation (3.25) shows that for adiabatic expansion
or, using (3.10)
55