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 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 $t_e=t_{0e}\Big(\frac{\rho t^3}{0.5(\rho t^3)_0}\Big)^a$, where $(\rho t^3)_0=\frac{0.05M_{\odot}}{4\pi(0.2c)^3}$, $t_{0e}(s_0,Y_e)\approx17$ days and $0.4\le a(s_0,Y_e)\le0.5$. The fractional uncertainty in $t_e$ due to nuclear physics uncertainties is $\approx10\%$. The result $a\le0.5$ 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 $50\%$ (typically $<30\%$, 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 (properly averaged) ejecta $\rho t^3$.