ner_tags: O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Formula I-Formula I-Formula O B-Model I-Model O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O B-Citation I-Citation O O O O O B-Citation I-Citation I-Citation O O O O O B-Citation I-Citation O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O B-Model I-Model I-Model O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Wavelength O O O O O
tokens: spectra are known in detail. Most importantly, geometrical effects cause the spectra to differ from a Planck spectrum. In particular, in the case of the photosphere occurring in the coasting phase, i.e. above the saturation radius, r s , the spectrum is significantly different from a Planck function (Goodman 1986 ; Beloborodov 2011 ; Lundman, Pe’er Ryde 2013 ). Its spectral shape is much broader, while the asymptotic, low-energy spectral slope is still very hard, with α ∼ 0.4. The most common way to assess different emission models in GRBs is by studying the sub-peak spectral shape. This is characterized by the photon index, α , of the Band function or, similarly, of the cut-off power-law function (Band, Matteson et al. 1993 ). The values of α can vary significantly depending on the emission process that is taking place. In an optically thick, thermal scenario, the Rayleigh–Jeans limit has α = 1, the Wien limit has α = 2, and the non-dissipative photosphere in the coasting phase has the expected α of 0.4. Non-thermal emission is always in the regime α 0. The limiting value can be reached by synchrotron emission in many specific cases, such as from electrons with a small pitch angle distribution (Lloyd Petrosian 2000 ), jitter radiation (Medvedev 2000 ), with attenuation by scattering (Dermer Böttcher 2000 ), and thermal synchrotron emission (Petrosian 1981 ). Likewise the limiting values of α = 0 can be reached by synchrotron self-Compton emission from monoenergetic electrons scattering off a self-absorbed seed photons field (Stern Poutanen 2004 ). More generally, instantaneous or fast synchrotron cooling can explain α −3/2 and slow synchrotron cooling is expected to occur at α −2/3 (e.g. Tavani 1996 ). However, before comparing predictions of physical emission models to the observed characteristics of bursts, such as the distribution of α values, limitations of the detector and the analysis methods must also be taken into account. Such limitations could cause spuriously deviation from the expected α (e.g. Preece et al. 1998 ; Lloyd Petrosian 2000 ). One such limitation is due to the limited band-width of the detector which prevents the full spectrum to be detected. Another limitation is due to the typically empirical models that are used to fit the data; the Band function and the cut-off power-law function (see e.g. Yu et al. 2016 ). If these empirical models do not match the true spectral shape, such as its curvature, then the fitted parameters might not be readily interpreted. There are two possible routes to address these limitations. One way is to fit physical models directly to the data, which eliminates the need for fits to empirical functions (e.g. Lloyd Petrosian 2000 ; Zhang Yan 2011 ; Ahlgren et al. 2015 ; Giannios 2006 ; Vianello et al. 2018 ; Burgess et al. 2018 ; Ahlgren et al. 2019 ). However, such analysis is computationally very demanding and, moreover, with the present, limited understanding of GRBs it is not clear what