An analytic derivation of the empirical correlations of gamma-ray bursts

¹ School of Astronomy and Space Science, Nanjing University, Nanjing 210023, PR China
e-mail: carpedieminreality@163.com; hyf@nju.edu.cn
² Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing, PR China
³ Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China
⁴ School of Cyber Science and Engineering, Qufu Normal University, Qufu 273165, PR China
⁵ College of Physics and Engineering, Qufu Normal University, Qufu 273165, PR China

Received: 9 November 2022
Accepted: 18 March 2023

Abstract

Empirical correlations between various key parameters have been extensively explored ever since the discovery of gamma-ray bursts (GRBs) and have been widely used as standard candles to probe the Universe. The Amati relation and the Yonetoku relation are two good examples that enjoyed special attention. The former reflects the connection between the peak photon energy (E_p) and the isotropic γ-ray energy release (E_iso), while the latter links E_p with the isotropic peak luminosity (L_p), both in the form of a power-law function. Most GRBs are found to follow these correlations well, but a theoretical interpretation is still lacking. Some obvious outliers may be off-axis GRBs and may follow correlations that are different from those at the on-axis. Here we present a simple analytical derivation for the Amati relation and the Yonetoku relation in the framework of the standard fireball model, the correctness of which is then confirmed by numerical simulations. The off-axis Amati relation and Yonetoku relation are also derived. They differ markedly from the corresponding on-axis relation. Our results reveal the intrinsic physics behind the radiation processes of GRBs, and they highlight the importance of the viewing angle in the empirical correlations of GRBs.