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C. Light Curves for the early radiative phase

If the electrons' energy is large (namely if epsilone is not far from unity), then early on during the first few hours of the afterglow there will be a radiative phase in which a significant fraction of the kinetic energy is lost via the radiative processes. One can generalize the BM solution to this radiative stage (see Cohen et al. [58] and Section VIIA). The essence of the radiative phase is that in this case the energy varies as E propto Gamma, where Gamma cong (R / L)-3. Note that L is calculated in terms of M and the initial energy of the explosion, E0, via M = E0 / Gamma0 c2, where Gamma0 is the initial Lorentz factor of the ejecta:

Equation 93 (93)

The transition time from the radiative to the adiabatic phase takes place when the radiation losses become negligible. This happens at:

Equation 94 (94)

Following Sari et al. [376] one can use the above expressions to express the different typical frequencies and fluxes as:

Equation 95 (95)

Like in the adiabatic case this can be translated to the times of passage of the break frequencies at a given observed frequency:

Equation 96 (96)

Unlike the adiabatic case, here nuc must be below num. Otherwise the bulk of the electrons do not cool and the system won't be radiative. Indeed at trad (given by Eq. 94 above) nuc = num.

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