An analytic description of electron thermalization in kilonovae ejecta

A simple analytic description is provided of the rate of energy deposition by β-decay electrons in the homologously expanding radioactive plasma ejected in neutron star mergers, valid for a wide range of ejecta parameters – initial entropy, electron fraction {s₀, Y_e}, and scaled (time-independent) density ρt³. The formulae are derived using detailed numerical calculations following the time-dependent composition and β-decay emission spectra (including the effect of delayed deposition). The deposition efficiency depends mainly on ρt³ and only weakly on {s₀, Y_e}. The time t_e at which the ratio between the rates of electron energy deposition and energy production drops to 1 − e⁻¹, is given by t_e = t_{0e (0.5(}^ρt_ρt³₃₎₀ )^a, where (ρt³)₀ = ₄⁰_π⁰⁵_(0.^M_{2c )3} , t_0e(s₀, Y_e) ≈ 17 d, and 0.4 ≤ a(s₀, Y_e) ≤ 0.5. The fractional uncertainty in t_e due to nuclear physics uncertainties is ≈ 10 per cent. The result a ≤ 0.5 reflects the fact that the characteristic β-decay electron energies do not decrease with time (largely due to ‘inverted decay chains’ in which a slowly decaying isotope decays to a rapidly decaying isotope with higher end-point energy). We provide an analytic approximation for the time-dependent electron energy deposition rate, reproducing the numerical results to better than 50 per cent (typically < 30 per cent, well within the energy production rate uncertainty due to nuclear physics uncertainties) over a 3–4 orders-of-magnitude deposition rate decrease with time. Our results may be easily incorporated in calculations of kilonovae light curves (with general density and composition structures), eliminating the need to numerically follow the time-dependent electron spectra. Identifying t_e, e.g. in the bolometric light curve, will constrain the ejecta density distribution.