An analytic description of electron thermalization in kilonovae ejecta

A simple analytic description is provided of the rate of energy deposition by beta-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 {s(0), Y-e}, and scaled (time-independent) density rho t(3). The formulae are derived using detailed numerical calculations following the time-dependent composition and beta-decay emission spectra (including the effect of delayed deposition). The deposition efficiency depends mainly on rho t(3) and only weakly on {s(0), 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) (rho t(3)/0.5(rho t(3))(0))(a), where (rho t(3))(0) =0.05M(circle dot)/4 pi(0.2c)(3), t(0e)(s(0), Y-e) approximate to 17 d, and 0.4 <= a(s(0), Y-e) <= 0.5. The fractional uncertainty in t(e) due to nuclear physics uncertainties is approximate to 10 per cent. The result a <= 0.5 reflects the fact that the characteristic beta-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.