\simeq 0.1\left( {\frac{\Gamma
_0}{10^2}} \right) ^2\left( {\frac{B_0}{10^3\,\rm{G}}}\right) \;\;
\rm{MeV}\,. 
\label{eps}$$

The average fraction of the kinetic energy of the beam that is radiated in these photons is

$$\xi _\gamma \equiv \frac{\left\langle \Phi _\gamma \right\rangle }{
n_0m_pc^3\Gamma _0}
\simeq 10^{-3}\left( {\frac{\Gamma _0}{10^2}}\right)
^2\left( {\frac{B_0}{10^3\,\rm{G}}}\right)\,,  
\label{ksinum}$$

where $\left\langle\Phi_\gamma \right\rangle$ is the average synchrotron luminosity of high-energy electrons per unit area of the barrier front.

In the burster frame, the characteristic energy, $\langle \tilde{\varepsilon}_
\gamma \rangle$, of synchrotron photons generated in the vicinity of the wind front increases due to the Doppler effect. Taking this into account and using equation (\[eps\]), we have

$$\langle \tilde{\varepsilon}_\gamma \rangle \simeq
10\left( {\frac{\Gamma _0 }{10^2}}\right) ^2\left(
{\frac{B_{\rm{dec}}}{10^5\,\rm{G}}}\right) \;\;\rm{ MeV}\,.
\label{epsLab}$$

where $B_{\rm{dec}}$ is determined in equation (\[Bdec\]). The fraction of the wind energy which is transferred to radiation at the wind front does not depend on the frame where it is estimated, and in the burster frame it is equal to

$$\tilde{\xi}_\gamma = \xi_\gamma\simeq  10^{-3}\left( \frac{\Gamma
_0}{10^2} \right) \left( \frac{B_{\rm{dec}}}{10^5\, \rm{G}}\right) \,,
\label{ksiLab}$$

For typical parameters of relativistic magnetized winds which are relevant to cosmological $\gamma$-ray bursters, $\Gamma_0\simeq 10^2-10^3$ and $B_{\rm{dec}}\simeq 10^5$ G (see § 2), from equations (\[epsLab\]) and (\[ksiLab\]) we have $\langle \tilde{\varepsilon}_\gamma \rangle\simeq 10 