3 The inverse solution  

Equations (1)-(4) are four independent relations among the four parameters of interest ${\cal E}_{52}$, n, $\epsilon_{\rm e}$, and $\epsilon_B$. This means we can solve for all parameters of interest if we have measured all three break frequencies (not necessarily at the same time) and the peak flux of the afterglow. In addition this requires us to know the redshift of the burst, the electron index p, and the composition parameter, X, of the ambient medium. We define the constants $C_{\rm a}\equiv\mbox{$\nu_{\rm a}$}/\mbox{$\nu_{\rm a}$}_*$,$C_{\rm m}\equiv\mbox{$\nu_{\rm m}$}\mbox{$t_{\rm d}$}_{\rm m}^{3/2}/\mbox{$\nu_{\rm m}$}_*$,$C_{\rm c}\equiv\mbox{$\nu_{\rm c}$}\mbox{$t_{\rm d}$}_{\rm c}^{1/2}/\mbox{$\nu_{\rm c}$}_*$, and $C_{\rm F}=F_{\mbox{$\nu_{\rm m}$}}/F_{\mbox{$\nu_{\rm m}$}*}$. Here starred symbols denote the numerical coefficients in each of the four equations, and times $\mbox{$t_{\rm d}$}_{\rm m}$, $\mbox{$t_{\rm d}$}_{\rm c}$ denote the time at which the quantity in question was measured. Rearranging the four equations then yields

$$\begin{eqnarray} \mbox{${\cal E}_{52}$}& = & \mbox{$C_{\rm a}$}^{-\frac{5}{6}} ... ...^{\frac{7}{2}} \left(\frac{\sqrt{1+z}-1}{h_{70}}\right)^{-3}\cdot\end{eqnarray}$$ (5) (6) (7) (8)