A Theory of Photospheric Emission From Relativistic Outflows

Abstract

Two popular models of optically thick relativistic outflows exist: the wind and the shell. We propose a unified treatment of photospheric emission within these models. We show that quite counterintuitive situations may appear when e.g. geometrically thin shell may behave as thick wind. For this reason we introduce notions of photon thick and photon thin outflows. They appear more general and better physically motivated than winds and shells when photospheric emission is considered.We obtain light curves and observed spectra for both photon thick and photon thin outflows. In the photon thick case we generalize the results obtained for steady wind. It is our main finding that the photospheric emission from the photon thin outflow is dominated by diffusion and produces non thermal time integrated spectra, which may be described by the Band function well known in the GRB literature.Energetic GRBs should produce photon thin outflows and therefore when only time integrated spectra for such GRBs are available we naturally expect them to have a Band shape. In the literature Band spectra for the photospheric emission of GRBs are obtained only involving additional dissipative mechanisms which are not required here.

Transparency Parameters From Relativistically Expanding Outflows

“…It was also found in Ruffini et al (2013) that radiative diffusion plays the main role in the photon decoupling when τ 0 4η 4 l/R 0 . Photons initially trapped in the outflow escape by random walk at the diffusion radius R D smaller than R ph :…”

“…The optical depth τ in such outflows depends on the activity duration of the central engine Δt measured in the burst restframe, in other words, the width of the outflow l cΔt. For finite outflows as considered here, Ruffini et al (2013) defines the photospheric radius R ph as the radius that fulfills the relation τ = 1 in the radial direction for a photon emitted at the inner most boundary of the outflow, see also Deng & Zhang (2014). R ph is given by the following:…”

Photospheric emission in gamma-ray bursts

“…11,189,190 However, full treatment of this effect for relativistic outflows, as occur in GRB, was carried out only recently. 5,185,[191][192][193][194][195] When considering spherical, relativistic explosion characterized by Γ ≫ 1, one can show that the photospheric radius is a strong function of the angle to the line of sight:…”

“…This realization led to the concept of a "vague photosphere" (See Figure 10). 5,128,191,[194][195][196][197] In a spherical explosion scenario, the effect of the vague photosphere on the observed spectral shape is not large; it somewhat modifies the Rayleigh-Jeans part of the spectrum, that reads F ν ∝ ν 3/2 (Ref. 185,196 ).…”

Photospheric emission in gamma-ray bursts

“…11,191,192 However, full treatment of this effect for relativistic outflows, as occur in GRB, was carried out only recently. 5,187,[193][194][195][196][197] Comptonization of nonisotropic distributed photons near the photosphere leads to substantial deviation from the "Planck" spectrum. 198,199 When considering spherical, relativistic explosion characterized by Γ 1, one can show that the photospheric radius is a strong function of the angle to the line of sight:…”