2 Generalization of the Band function

The "Band" function (Band et al. 1993) is a four-parameter phenomenological fit consisting of two smoothly connected power laws that is widely used to fit GRB spectra. Typically, the rising $\nu F_\nu$ indices $\u$ below the peak $\nu F_\nu$ energy ${\epsilon}_{\rm peak}$have measured values $\u \cong 1$,and the falling indices $\d$ have values $\d \approx 0-0.5$, where $\u ,\d \gt$by definition. The prediction of the nonthermal synchrotron shock model is that $\u \leq 4/3$ as a result of the intrinsic synchrotron emissivity spectrum.

We have proposed (Dermer et al. 1999a) a temporally-evolving generalization of the Band form based on the physics of the blast wave model. This formula is essentially a Band function with both the amplitude and ${\epsilon}_{\rm peak}$ changing with time. Nine parameters in all are necessary to define a model, namely the (i) total energy E₀ of the explosion; (ii) $\Gamma_0$; (iii) the equipartition factor q; (iv) the density n₀ and (v) radial-gradient index $\eta$ of the CBM (see below); (vi) the redshift z of the source; (vii) the radiative regime g describing the blast wave deceleration $\Gamma \propto x^{-g}$; and the spectral indices (viii) $\u$ and (ix) $\delta$. We also assume that q, $\u$, and $\delta$ are independent of time. This formula lacks self-consistency in the sense that a magnetic field of sufficient strength must be present for the blast wave to be found in a particular radiative regime. These limitations of the model can only be addressed through detailed numerical simulations (e.g., Chiang & Dermer 1999).