Radiation diffusion in a ultra-relativistic expanding shell in relation to gamma-ray bursts

 The present-day observational data obtained by satellite  observatories cover seven decades of gamma-ray energy, and there is no universal general model describing the formation of the spectrum. Therefore, it is important to describe the initial stages of radiation propagation in an ultrarelativistically expanding shell. The aim of this study was to obtain equations describing the propagation of radiation in a relativistically expanding shell in the diffusion limit, solve them for natural initial data, and apply the results obtained to the initial radiation of gamma-ray bursts. The following results were obtained: the initial stage of the gamma-ray burst in a photon-thin case can be described by radiation diffusion in an ultrarelativistically expanding shell; the time interval at which it is still possible to use the diffusion approximation increases with increasing the depth inside the shell quadratically; the value of the depth beyond which the diffusion approximation can be used increases, and the value of the radiation intensity decreases in diffusion time approaches; during the main radiation of the photon-thin shell, the diffusion approximation is suitable for most of the jet. The parameters of emission are close to the ones of short gamma-ray bursts.

1. Bisnovatyi-Kogan G. S. Cosmic gamma-ray bursts: observations and simulations. Fizika elementarnyh chastic i atomnogo yadra = Physics of Elementary Particles and Atomic Nuclei, 2006, vol. 37, no. 5, pp. 1236−1284 (in Russian).

2. Kumar P., Zhang B. The physics of gamma-ray bursts & relativistic jets. Physics Reports, 2015, vol. 561, pp. 1–109. https://doi.org/10.1016/j.physrep.2014.09.008

3. Piran T. The The physics of gamma-ray bursts. Reviews of Modern Physics, 2005, vol. 76, no. 4, pp. 1143–1210. https:// doi.org/10.1103/revmodphys.76.1143

4. Biscoveanu S., Thrane E., Vitale S. Constraining short gamma-ray burst jet properties with gravitational waves and gamma rays. The Astrophysical Journal, 2020, vol. 893, no. 1, pp. 38. https://doi.org/10.3847/1538-4357/ab7eaf

5. Haskell R. C., Svaasand L. O., Tsong-Tseh Tsay, Ti-Chen Feng, Tromberg B. J., McAdams M. S. Boundary condi tions for the diffusion equation in radiative transfer. Journal of the Optical Society of America A, 1994, vol. 11, no. 10, pp. 2727−2741. https://doi.org/10.1364/josaa.11.002727

6. Zhang B. The Physics of Gamma-Ray Bursts. Reviews of Modern Physics, 2004, vol. 76, no. 4. pp. 1143−1210. https:// doi.org/10.1103/revmodphys.76.1143

7. Pe’er А. Physics of Gamma-Ray Bursts Prompt Emission. Advances in Astronomy, 2015, vol. 2015, pp. 1–37. https://doi. org/10.1155/2015/907321

8. Abdalla H., Adam R., Aharonian F. A very-high-energy component deep in the γ-ray burst afterglow. Nature, 2019, vol. 575, pp. 464−467. https://doi.org/10.1038/s41586-019-1743-9

9. Zhang B., Mészáros P. Gamma-Ray Bursts: progress, problems & prospects. International Journal of Modern Physics, 2004, vol. 19, no. 15, pp. 2385−2472. https://doi.org/10.1142/s0217751x0401746x

10. Mészáros P. Gamma-ray bursts. Reports on Progress in Physics, 2006, vol. 69, no. 8, pp. 2259−2321. https://doi. org/10.1088/0034-4885/69/8/r01

11. Kulkarni S., Desai S. Classification of gamma-ray burst durations using robust model-comparison techniques. Astrophysics and Space Science, 2017, vol. 362, no. 4. https://doi.org/10.1007/s10509-017-3047-6

12. Fox D. B., Frail D. A., Price P. A., Kulkarni S. R., Berger E., Piran T., Soderberg A. M., Cenko S. B., Cameron P. B., Gal-Yam A., Kasliwal M. M., Moon D.-S., Harrison F. A., Nakar E., Schmidt B. P., Penprase B., Chevalier R. A. [et al.]. The afterglow of GRB 050709 and the nature of the short-hard γ-ray bursts. Nature, 2005, vol. 437, no. 7060, pp. 845−850. https:// doi.org/10.1038/nature04189

13. Ruffini R., Siutsou I. A., Vereshchagin G. V. A theory of photospheric emission from relativistic outflows. The Astrophysical Journal, 2013, vol. 772, no. 1, pp. 11. https://doi.org/10.1088/0004-637x/772/1/11

14. Rybicki G. B., Lightman A. P. Radiative Processes in Astrophysics. Weinheim, WILEY-VCH Verlag GmbH & Co. KGaA, 2004, 382 p. https://doi.org/10.1002/9783527618170

15. Beloborodov А. М. Radiative Transfer in Ultrarelativistic Outflows. The Astrophysical Journal, 2011, vol. 737, no. 2, pp. 68. https://doi.org/10.1088/0004-637x/737/2/68

16. Mihalas D. Solution of the comoving-frame equation of transfer in spherically symmetric flows. VI - Relativistic flows. The Astrophysical Journal, 1980, vol. 237, pp. 574−589. https://doi.org/10.1086/157902

17. Vereshchagin G. V., Siutsou I. A. Diffusive photospheres in gamma-ray bursts. Monthly Notices of the Royal Astronomical Society, 2020, vol. 494, no. 1, pp. 1463−1469. https://doi.org/10.1093/mnras/staa868