2 Synchrotron radiation from an adiabatic blast wave  

We have derived the synchrotron frequencies and fluxes for a relativistic blast wave expanding in a uniform medium. The results for the peak flux, $F_{\nu_{\rm m}}$, the synchrotron self-absorption, $\nu_{\rm a}$, peak, $\nu_{\rm m}$, and cooling, $\nu_{\rm c}$,frequencies are summarised below (details can be found in Wijers & Galama 1999).

$$\begin{eqnarray} & F_{\mbox{$\nu_{\rm m}$}} \! =\! 1.15 \frac{h_{70}^2}{(\sqrt{... ...^{-1/2} \mbox{$t_{\rm d}$}^{-1/2}\:\mbox{\rm\thinspace Hz$^{}$}.\end{eqnarray}$$ (1) (2) (3) (4)

Here $x_{\rm p}$ and $\phi_{\rm p}$ are the dimensionless spectral location and dimensionless peak flux of a synchrotron spectrum from a power law distribution of electron energies with index p. X is the hydrogen mass fraction, $\mbox{$\epsilon_{\rm e}$}$ and $\mbox{$\epsilon_B$}$ are the ratios of energy in electrons and energy in the magnetic field to energy in nucleons, respectively, ${\cal E}_{52}$ is the blast wave energy per unit solid angle in units of 10⁵² erg, n is the nucleon density, h₇₀= H₀/70 km s^-1 Mpc^-1, z the redshift and $t_{\rm d}$ the time in days since trigger (see Wijers & Galama 1999). We have adopted an $\Omega=1, \Lambda=0$ universe.