8.3. Synchrotron Self Absorption

An important effect that we have ignored so far is the possibility of self absorption. This is irrelevant during the GRB itself. One of the essential features of the GRB spectrum is that it is produced in an optically thin region. However, self absorption may appear at late time and typically in radio emission [18, 247, 25, 248, 249]. When it appears it will cause a steep cutoff of the low energy spectrum, either as the commonly known ^5/2 or as ².

To estimate the self absorption frequency we need the optical depth along the line of sight. A simple approximation is: '_' R_l / ₁ where '_' is the absorption coefficient [245]:

(67)

The self absorption frequency _a satisfies: '_'0 R / = 1. It can be estimates only once we have a model for the hydrodynamics and how do R and change with time [248, 249].

The spectrum below the the self-absorption frequency depends on the electron distribution. One obtaine the well known [245], ^5/2 slop when the synchrotron frequency of the electron emitting the self absorbed radiation is inside the self absorption range. One obtains a slope of ²2 if there is self absorption, but the radiation in that range is due to the low energy tail of electrons radiating effectively at higher energies. For this latter case, which is more appropriate for GRB afterglow we find that [18, 25]:

(68)

where R is the radius of the radiating shell and the factor k_B T_e / ( m_p c²) describes the degree of electron equipartition in the plasma shock-heated to an internal energy per particle m_p c² and moving with Lorentz factor .