Low Energy Spectral Index and $E_{p}$ evolution of Quasi-thermal Photosphere Emission of Gamma-Ray Bursts

Abstract

Recent observations by the Fermi satellite suggest that a photosphere emission component is contributing to the observed spectrum of many GRBs. One important question is whether the photosphere component can interpret the typical ``Band'' function of GRBs with a typical low energy photon spectral index $\alpha \sim -1$. We perform a detailed study of the photosphere emission spectrum by progressively introducing several physical ingredients previously not fully incorporated, including the probability distribution of the location of a dynamically evolving photosphere, superposition of emission from an equal-arrival-time ``volume'' in a continuous wind, the evolution of optical depth of a wind with finite but evolving outer boundary, as well as the effect of different top-hat wind luminosity ($L_w$) profiles. By assuming a co-moving blackbody spectrum emerging from the photosphere, we find that for an outflow with a constant or increasing $L_w$, the low-energy spectrum below the peak energy ($E_{p}$), can be modified to $F_\nu \sim \nu^{1.5}$ ($\alpha \sim +0.5$). A softer ($-1<\alpha<+0.5$) or flat ($\alpha=-1$) spectrum can be obtained during the $L_w$ decreasing phase or high-latitude-emission-dominated phase. We also study the evolution of $E_{p}$ as a function of wind and photosphere luminosity in this photosphere model. An $E_p-L$ tracking pattern can be reproduced if a certain positive dependence between the dimensionless entropy $\eta$ and $L_w$ is introduced. However, the hard-to-soft evolution pattern cannot be reproduced unless a contrived condition is invoked. In order to interpret the Band spectrum, a more complicated photosphere model or a different energy dissipation and radiation mechanism are needed.