Photosphere emission from a hybrid relativistic outflow with arbitrary dimensionless entropy and magnetization in GRBs

In view of the recent Fermi observations of GRB prompt emission spectra, we develop a theory of photosphere emission of a hybrid relativistic outflow with a hot fireball component (defined by dimensionless entropy $\eta$) and a cold Poynting-flux component (defined by magnetization $\sigma_0$ at the central engine). We consider the scenarios both without and with sub-photospheric magnetic dissipations. Based on a simplified toy model of jet dynamics, we develop two approaches: a "bottom-up" approach to predict the temperature (for a non-dissipative photosphere) and luminosity of the photosphere emission and its relative brightness for a given pair of $(\eta,\sigma_0)$; and a "top-down" approach to diagnose central engine parameters ($\eta$ and $\sigma_0$) based on the observed quasi-thermal photosphere emission properties. We show that a variety of observed GRB prompt emission spectra with different degrees of photosphere thermal emission can be reproduced by varying $\eta$ and $\sigma_0$ within the non-dissipative photosphere scenario. In order to reproduce the observed spectra, the outflows of most GRBs need to have a significant $\sigma$, both at the central engine, and at the photosphere. The $\sigma$ value at $10^{15}$ cm from the central engine (a possible non-thermal emission site) is usually also greater than unity, so that internal-collision-induced magnetic reconnection and turbulence (ICMART) may be the mechanism to power the non-thermal emission. We apply our top-down approach to GRB 110721A, and find that the temporal evolution behavior of its blackbody component can be well interpreted with a time-varying $(\eta,\sigma_0)$ at the central engine, instead of invoking a varying engine base size $r_0$ as proposed by previous authors.