On the $\alpha$-Intensity Correlation in Gamma-Ray Bursts: Subphotospheric Heating with Varying Entropy

The emission mechanism during the prompt phase in gamma-ray bursts (GRBs) can be investigated through correlations between spectral properties. Here, we revisit the correlation relating the instantaneous flux, $F$, and the photon index below the spectral break, $\alpha$, in individual emission pulses, by studying the 38 most prominent pulses in the Fermi/GBM GRB catalogue. First, we search for signatures of the bias in the determination of $\alpha$ due to the limited spectral coverage (window effect) expected in the synchrotron case. The absence of such a characteristic signature argues against the simplest synchrotron models. We instead find that the observed correlation between $F$ and $\alpha$ can, in general, be described by the relation $F(t) \propto {\rm e}^{k\,\alpha(t)}$, for which the median $k = 3$. We suggest that this correlation is a manifestation of subphotospheric heating in a flow with a varying entropy. Around the peak of the light curve, a large entropy causes the photosphere to approach the saturation radius, leading to an intense emission with a narrow spectrum. As the entropy decreases the photosphere secedes from the saturation radius, and weaker emission with a broader spectrum is expected. This simple scenario naturally leads to a correlated variation of the intensity and spectral shape, covering the observed range.