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6.3. The Radiation-Dominated Phase

Initially the fireball is radiation dominated. For gamma << (e0 / rho0) gamma0, the first term in Eq 20 dominates and we find D propto r, gamma propto r, recovering the radiation-dominated scaling:

Equation 22 (22)

The scalings of rho 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 Deltar ~ r / gamma ~ constant ~ Ri, 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 partial / partials ~ gamma / 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 / gamma. Therefore, the conservation laws 19 and the scalings 22 are valid so long as the radiation-dominated fireball expands ultra-relativistically with large gamma. The only possible exception is in the very outermost layers of the fireball where the pressure gradient may be extremely steep and partial / partials may be >> gamma / r. Ignoring this minor deviation, we interpret Eq. 19 and the constancy of the radial width Deltar 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.

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