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, Γ=Γ(r, Θ, φ), where Γ 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 Γ ≈ Γ0/[(Θ/Θ_j)^p + 1], where Θj is the characteristic jet opening angle. In collimated jets, the observed spectrum depends on the viewing angle, Θv. We show that for narrow jets (Θ_jΓ0 < few), the obtained low-energy photon index is α ≈-1 (dN/dE / Eα), independent of viewing angle, and weakly dependent on the Lorentz factor gradient (p). A similar result is obtained for wider jets observed at Θv ≈ Θj. This result is surprisingly similar to the average low-energy photon index seen in gamma-ray bursts. For wide jets (Θ_jΓ0 > few) observed at Θv > Θj, a multicolour blackbody spectrum is obtained. We discuss the consequences of this theory on our understanding of the prompt emission in gamma-ray bursts.