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6.6. Optical Depth

Independently of the above considerations, at some point during the expansion, the fireball will become optically thin. For a pure fireball this happens when the local temperature drops to about 20 keV at:

Equation 26 (26)

In a matter dominated fireball the optical depth is usually determined by the ambient electrons. In this case the fireball becomes optically thin at:

Equation 27 (27)

From this stage on the radiation and the baryons no longer move with the same velocity and the radiation pressure vanishes, leading to a breakdown of Eqs. 16-18. Any remaining radiation will escape freely now. The baryon shells will coast with their own individual velocities. If the fireball is already in the matter dominated coasting phase there will be no change in the propagation of the baryons. However, if the fireball is in the radiation dominated phase when it becomes optically thin, then the baryons will switch immediately to a coasting phase. This transition radius, Re has another crucial role in the fireball evolution. It is the minimal radius in which energy conversion and generation of the observed GRB can begin. Photons produced at R < Re cannot escape.