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6.4. The Matter-Dominated Phase

The radiation dominated regime extends out to a radius r ~ (e0 / rho0) r0. At larger radii, the first and last terms in Eq. 20 become comparable and gamma tends to its asymptotic value of gammaf = (4e0 / 3rho0 + 1)gamma0. This is the matter dominated regime. (The transition occurs when 4e / 3 = rho, which happens when gamma = gammaf / 2.) In this regime, D propto r2/3, leading to the scalings:

Equation 23 (23)

The modified scalings of rho and e arise because the fireball now moves with a constant radial width in the comoving frame. (The steeper fall-off of e with r is because of the work done by the radiation through tangential expansion.) Moreover, since e << rho, the radiation has no important dynamical effect on the motion and produces no significant radial acceleration. Therefore, gamma remains constant on streamlines and the fluid coasts with a constant asymptotic radial velocity. Of course, since each shell moves with a velocity that is slightly less than c and that is different from one shell to the next, the frozen pulse approximation on which Eq. 19 is based must ultimately break down at some large radius.