**6.3. The Radiation-Dominated Phase**

Initially the fireball is radiation dominated. For
<<
(*e*_{0} /
_{0})
_{0},
the first term in Eq 20 dominates and we find
*D* *r*,
*r*, recovering
the radiation-dominated scaling:

(22) |

The scalings of
and *e*
given in Eq. 22 correspond to
those of a fluid expanding uniformly in the comoving frame. Indeed,
all three scalings in Eq. 22 can be derived for a homogeneous
radiation dominated fireball by noting the analogy with an expanding
universe.

Although the fluid is approximately homogeneous in its own frame,
because of Lorentz contraction it appears as a narrow shell in the
observer frame, with a radial width given by
*r* ~ *r*
/ ~
constant ~ *R*_{i}, the initial radius of the fireball, or
the initial width of the specific shell under discussion when we consider
a continuous wind. We can now go back to Eqs. 16-18 and
set /
*s* ~
/
*r*. We then find that the terms
we neglected on the right hand sides of these equations are smaller
than the terms on the left by a factor
~ 1 / .
Therefore, the conservation laws 19 and the scalings 22 are valid so long
as the radiation-dominated fireball expands ultra-relativistically
with large .
The only possible exception is in the very outermost layers of the
fireball where the pressure gradient may be extremely steep and
/
*s* may be
>>
/ *r*. Ignoring this minor deviation, we interpret Eq. 19 and the
constancy of the radial width
*r* in the
observer frame to mean
that the fireball behaves like a pulse of energy with a frozen radial
profile, accelerating outward at almost the speed of light.