Abstract
The Combo-relation is a tight relation involving the prompt emission spectral peak energy Ep,i and the X-ray afterglow plateau luminosity L0 and its rest-frame duration τ, and the late power-law decay index α. To explain its features, in this work we investigate the external shock scenario, i.e. the synchrotron emission from the electrons accelerated in a relativistic shock propagating through a cold external medium with density n and interacting with the magnetic field B associated with the turbulent plasma. Three assumptions were introduced: (1) the homologous shock expansion; (2) an electron power-law distribution over a large range of frequencies; and (3) a power-law dependence on the shock radius of B and n. First, we show how the values of Ep,i, L0 and τ depend on combinations of the initial values of the magnetic field B0, shock radius R0 and medium density n0, as well as the shock Lorentz factor Γ0 and the electron distribution. Then, we prove that the Combo-relation results from the relationship between Ep,i, L0 and τ and the corresponding comoving quantities that scale with Γ0. Finally, through Monte Carlo simulations, we deduce the typical model parameters, for example, |$B_0=4.6^{+253.9}_{-4.5}$| G, |$R_0=7.0^{+34.4}_{-5.8}\times 10^{13}$| cm and |$n_0=10.00^{+7694.62}_{-9.99}$| cm−3, all in agreement with the results obtained in the literature. We conclude that, within the external shock scenario and its flaws, the Combo-relation represents a new tool for investigating the physics of the prompt and afterglow emissions and for inferring the properties of the gamma-ray-burst external medium.