In view of the recent Fermi observations of GRB prompt emission spectra, wedevelop a theory of photosphere emission of a hybrid relativistic outflow witha hot fireball component (defined by dimensionless entropy $\eta$) and a coldPoynting-flux component (defined by magnetization $\sigma_0$ at the centralengine). We consider the scenarios both without and with sub-photosphericmagnetic dissipations. Based on a simplified toy model of jet dynamics, wedevelop two approaches: a "bottom-up" approach to predict the temperature (fora non-dissipative photosphere) and luminosity of the photosphere emission andits 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 onthe observed quasi-thermal photosphere emission properties. We show that avariety of observed GRB prompt emission spectra with different degrees ofphotosphere thermal emission can be reproduced by varying $\eta$ and $\sigma_0$within the non-dissipative photosphere scenario. In order to reproduce theobserved 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 emissionsite) is usually also greater than unity, so that internal-collision-inducedmagnetic reconnection and turbulence (ICMART) may be the mechanism to power thenon-thermal emission. We apply our top-down approach to GRB 110721A, and findthat the temporal evolution behavior of its blackbody component can be wellinterpreted with a time-varying $(\eta,\sigma_0)$ at the central engine,instead of invoking a varying engine base size $r_0$ as proposed by previousauthors.
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