**9.2. Phases in a Relativistic Decelerating Shell**

There are several phases in the deceleration of a relativistic shell:
fast cooling (with either radiative or adiabatic hydrodynamics) is
followed by slow cooling (with adiabatic hydrodynamics). Then if the
shell is non spherical its evolution changes and a phase of sideways
expansion and much faster slow down begins when the Lorentz factor
reaches ^{-1}
[258].
Finally the shell becomes Newtonian when enough mass is collected and
1. In the
following we estimate the time scale for the different transitions. We
define _{e,min}
*c*_{}
_{e}(*m*_{p} / *m*_{e})
and
*t*_{obs} = (1 + *z*) *R* / 4 *c*_{t}
*c*
^{2}
such that the factors
*c*_{} and *c*_{t} reflect some of the
uncertainties in the model. The canonical values of these factors are:
*c*_{}
0.5 and
*c*_{t} 1.

The deceleration begins in a fast cooling phase. If
_{e} is
close to unity than this cooling phase will also be radiative. The
first transition is from fast to slow cooling.
There are several different ways to estimate this transition.
One can compare the cooling time scale to the hydrodynamic time scale;
alternatively one can calculate the fast cooling rate (given by the
rate of energy generation by the shell)
and compare it to the slow cooling rate (given by the emissivity of
the relativistic electrons). We have chosen here to calculate this
time as the time when the "typical electron" cools - that is
when _{c} =
_{m}:

(126) |

All methods of estimating *t*_{fs} give the same dependence on
the parameters. However, the numerical factor is quite sensitive to
the definition of this transition.

If the solution is initially radiative the transition from fast to slow cooling and from a radiative hydrodynamics to adiabatic hydrodynamics takes place at:

(127) |

During a radiative evolution the energy in the shock decreases with
time. The energy that appears in Eqs. 122 in the radiative
scalings is the initial energy. When a radiative shock switches to
adiabatic evolution, it is necessary to use the reduced energy to
calculate the subsequent adiabatic evolution. The energy *E*_{f,
52}
which one should use in the adiabatic regime is related to the initial
*E*_{i, 52} of the fireball by

(128) |

If the shell is not spherical and it has an opening angle:
,
then the evolution will change when
~
^{-1}
[258].
Earlier on the jet expands too rapidly to expand sideways
and it evolves as if it is a part of a spherical shell. After this
stage the jet expands sideways and it accumulates much more mass and
slows down much faster. This transition will take place, quite
generally, during the adiabatic phase at:

(129) |

The shell eventually becomes non relativistic. This happens at:
*R* *l*
= (4*E*_{0} / 4
*n*_{1} *m*_{p}
*c*^{2})^{1/3} for an adiabatic
solution. This corresponds to a transition at:

(130) |

A radiative shell loses energy faster and it becomes non relativistic
at *R* = *L* = *l* /
_{0}^{1/3} = (4 *E*_{0} /
4 *n*_{ism}
*m*_{p} *c*^{2}
_{0})^{1/3}. This will take place at:

(131) |

However, the earlier estimate of the transition from fast to slow cooling suggests that the shell cannot remain radiative for such a long time.