Theory of the Fireball
Theory of the Fireball
Theory of the Fireball
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6. Sinrinkage <strong>of</strong> Isotnemal Sphere<br />
We denote <strong>the</strong> 'tmass'' <strong>of</strong> <strong>the</strong> iso<strong>the</strong>rmal sphere, as defined by (6.1)~<br />
by 5, Then this mass will decrease, due to <strong>the</strong> progress <strong>of</strong> tne coding<br />
wave inward, according to<br />
d"l<br />
dt= - PO Rf<br />
Note <strong>the</strong> po 5n <strong>the</strong> denominator and <strong>the</strong> absence <strong>of</strong> <strong>the</strong> factor &fly both<br />
due to <strong>the</strong> definition (6.1) The density <strong>of</strong> <strong>the</strong> iso<strong>the</strong>rmal sphere is<br />
nearly uniform and vi11 be denoted by p<br />
is<br />
Initial values at time tar<br />
will be denoted by a subscript a. The iso<strong>the</strong>rmal sphere expands adia-<br />
bat ically, hence<br />
and <strong>the</strong>refore at any time t<br />
The speed <strong>of</strong> <strong>the</strong> cooling wave is given by (5 .LO), thus<br />
u=<br />
5<br />
- Hl<br />
(6.4)<br />
where J has been set equal to zero because <strong>the</strong>re is no appreciable flow<br />
0