# Gamma-ray Bursts: 50 Years and Counting!
## 3 The BATSE Era (1991-1996)
\begin{cases}A\left(\frac{E}{100\,\text{keV}}\right)^{ \alpha}\exp\left(-\frac{E(2+\alpha)}{E_{\text{peak}}}\right)&E<E_{c}\\ A\left[\frac{(\alpha-\beta)E_{\text{peak}}}{100\,\text{keV}(2+\alpha)}\right]^ {\alpha-\beta}\exp(\beta-\alpha)\left(\frac{E}{100\,\text{keV}}\right)^{\beta }&E\geq E_{c}\end{cases} , \tag{2}\]  
where  
\[E_{c}=(\alpha-\beta)\frac{E_{\text{peak}}}{2+\alpha}\equiv(\alpha-\beta)E_{0}\]  
\(\alpha\) is the low-energy index, \(\beta\) is the high-energy index, \(E_{c}\) is the break energy, and \(E_{\text{peak}}\) is the peak energy of the \(vF_{v}\) spectrum. No specific theoretical model predicts this spectral shape; however, the Band function provides an excellent fit to most observed spectra. Most GRBs show substantial spectral evolution with two typical behaviors: hard to soft throughout the whole GRB duration or hard to soft during each pulse. The value of \(E_{\text{peak}}\) generally follows the shape of the light curve. In the same year, CGRO/EGRET detected high-energy (HE) photons, from 30 MeV up to 18 GeV, for a few GRBs, with emission lasting up to thousands of seconds after the onset of the GRB. The average spectrum of the events detected by EGRET is well fitted by a single power law with an index of \(\sim\)2, consistent with the extension of the low-energy spectra. While the exact distances of GRB sources remained a topic of discussion, the data collected by BATSE offered pivotal insights strongly indicative of the cosmological nature of GRBs. GRB signals were found to be distributed isotropically in the sky, with no clustering near the Galactic Disk (as predicted by galactic models) [19]. However, this did not completely rule out the galactic models, which were recalibrated to shift the GRB sources to the galactic halo to smooth out the inconsistency with the observed distribution. The intensity distribution of the bursts was found to deviate from homogeneity at the faint end, indicating a deficit of weak bursts compared to the predictions [20]. The intensity distribution provides information on the radial distribution of the sources and is defined as the ratio \(V/V_{\text{max}}=(C_{\text{max}}/C_{\text{min}})^{-3/2}\), where \(V\) is the volume contained by a sphere extending to the location of the burst, \(V_{\text{max}}\) is the volume of a sphere extending to the maximum distance at which the burst would still be detectable by the instrument, \(C_{\text{max}}\) is the peak count rate, \(C_{\text{min}}\) is the minimum rate required for triggering, and \(-3/2\) is the power-law exponent predicted if the burst follows a homogeneous distribution in Euclidean space. For a homogeneous distribution of sources, the average ratio would be \(\langle V/V_{\text{max}}\rangle=1/2\). BATSE reported an average ratio of \(\langle V/V_{\text{max}}\rangle=0.348\pm 0.024\), showing a deficit of low-fluence bursts and strongly hinting at a deviation from spatial homogeneity of the distribution of the sources. These findings imposed great constraints on current galactic models but could be easily explained if GRBs have an extragalactic origin. Despite great efforts to locate extragalactic GRBs with a certain precision, both through BATSE module counts and through temporal triangulation with other instruments(BATSE Gamma-ray Burst Coordinates Distribution Network, BACODINE, now known as the GCN-GRB Coordinates Network), no host galaxy was identified, which is the so-called \"No host problem\". During this era, the extragalactic models received a huge push in development. [PERSON] and [PERSON] proposed, in a series of papers, the now-standard fireball shock model, which already included the main ingredients of the current GRB theoretical framework, from the external shock of a relativistic fireball and synchrotron radiation as the main radiation mechanism to the external reverse shock and inverse Compton scattering.