6.1. A simple model
The evolution of a homogeneous fireball can be understood by a simple analogy to the Early Universe . Consider, first, a pure radiation fireball. If the initial temperature is high enough pairs will form. Because of the opacity due to pairs, the radiation cannot escape. The pairs-radiation plasma behaves like a perfect fluid with an equation of state p = / 3. The fluid expands under of its own pressure. As it expands it cools with R-1 ( being the local temperature and R the radius). The system resembles quite well a part of a Milne Universe in which gravity is ignored. As the temperature drops below the pair-production threshold the pairs annihilate. When the local temperature is around 20 keV the number of pairs becomes sufficiently small, the plasma becomes transparent and the photons escape freely to infinity. In the meantime the fireball was accelerated and it is expanding relativistically outwards. Energy conservation (as viewed from the observer frame) requires that the Lorentz factor that corresponds to this outward motion satisfies R. The escaping photons, whose local energy (relative to the fireball's rest frame) is 20 keV are blue shifted. An observer at rest detects them with a temperature of obs . Since R-1 and R we find that the observed temperature, obs, approximately equals 0, the initial temperature. The observed spectrum, is however, almost thermal  and it is still far from the one observed in GRBs.
In addition to radiation and e+ e- pairs, astrophysical fireballs may also include some baryonic matter which may be injected with the original radiation or may be present in an atmosphere surrounding the initial explosion. These baryons can influence the fireball evolution in two ways. The electrons associated with this matter increase the opacity, delaying the escape of radiation. Initially, when the local temperature is large, the opacity is dominated by e+ e- pairs . This opacity, p, decreases exponentially with decreasing temperature, and falls to unity when = p 20 keV. The matter opacity, b, on the other hand decreases only as R-2, where R is the radius of the fireball. If at the point where p = 1, b is still > 1, then the final transition to = 1 is delayed and occurs at a cooler temperature.
More importantly, the baryons are accelerated with the rest of the fireball and convert part of the radiation energy into bulk kinetic energy. The expanding fireball has two basic phases: a radiation dominated phase and a matter dominated phase. Initially, during the radiation dominated phase the fluid accelerates with R. The fireball is roughly homogeneous in its local rest frame but due to the Lorentz contraction its width in the observer frame is Ri, the initial size of the fireball. A transition to the matter dominated phase takes place when the fireball has a size
and the mean Lorentz factor of the fireball is E / M c2. We have defined here E52 E / 1052 ergs and Ri7 Ri / 107 cm. After that, all the energy is in the kinetic energy of the matter, and the matter coasts asymptotically with a constant Lorentz factor.
The matter dominated phase is itself further divided into two sub-phases. At first, there is a frozen-coasting phase in which the fireball expands as a shell of fixed radial width in its own local frame, with a width ~ Ri ~ (E / M c2)Ri. Because of Lorentz contraction the shell appears to an observer with a width Ri. Eventually, when the size of the fireball reaches Rs = 2 1011 cm( / 107 cm)( / 100)2 variability in within the fireball results in a spreading of the fireball which enters the coasting-expanding phase. In this final phase, the width of the shell grows linearly with the size of the shell, R:
The initial energy to mass ratio, = (E / M c2), determines the order of these transitions. There are two critical values for :
These correspond to four different types of fireballs:
|Type||= E / M c2||M|
|Pure Radiation||pair <||M < Mpair = 10-12 M E521/2 Ri71/2|
|Electrons Opacity||b < < pair||Mpair < M < Mb = 2 . 10-7 M E522/3 Ri72/3|
|Relativistic Baryons||1 < < b||Mb < M < 5 . 10-3 M E52|
|Newtonian||< 1||5 . 10-4 M E52 < M|
(i) A Pure Radiation Fireball (pair < ): The effect of the baryons is negligible and the evolution is of a pure photon-lepton fireball. When the temperature reaches p, the pair opacity p drops to 1 and b << 1. At this point the fireball is radiation dominated (E > M c2) and most of the energy escapes as radiation.
(ii) Electron Dominated Opacity (b < < pair): In the late stages, the opacity is dominated by free electrons associated with the baryons. The comoving temperature decreases far below p before reaches unity. However, the fireball continues to be radiation dominated and most of the energy still escapes as radiation.
(iii) Relativistic Baryonic Fireball (1 < < b): The fireball becomes matter dominated before it becomes optically thin. Most of the initial energy is converted into bulk kinetic energy of the baryons, with a final Lorentz factor f (E / M c2). This is the most interesting situation for GRBs.
(iv) Newtonian Fireball ( < 1): This is the Newtonian regime. The rest energy exceeds the radiation energy and the expansion never becomes relativistic. This is the situation, for example in supernova explosions in which the energy is deposited into a massive envelope.