2017MNRAS.468..570M

The Combo-relation is a tight relation involving the prompt emission spectral peak energy E_p,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 E_p,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 E_p,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^_–5.8×10¹³ 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.