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A theory of photospheric emission from relativistic, collimated outflows

Relativistic outflows in the form of jets are common in many astrophysical objects. By their very nature, jets have angle dependent velocity profiles, Gamma = Gamma(r, theta, phi), where Gamma is the outflow Lorentz factor. In this work we consider photospheric emission from non-dissipative jets with various Lorentz factor profiles, of the approximate form Gamma \approx Gamma_0/[(theta/theta_j)^p + 1], were theta_j is the characteristic jet opening angle. In collimated jets, the observed spectrum depends on the viewing angle, theta_v. We show that for narrow jets (theta_j Gamma_0 \lesssim few), the obtained low energy photon index is alpha \approx -1 (dN/dE \propto E^alpha), independent of viewing angle, and weakly dependent on the Lorentz factor gradient (p). A similar result is obtained for wider jets observed at theta_v \approx theta_j. This result is surprisingly similar to the average low energy photon index seen in gamma-ray bursts. For wide jets (theta_j Gamma_0 \gtrsim few) observed at theta_v \ll theta_j, a multicolor blackbody spectrum is obtained. We discuss the consequences of this theory on our understanding of the prompt emission in gamma-ray bursts.