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/θj)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 (θjΓ0 ≲ 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 (θjΓ0 ≳ 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.