Electron density as a function of time for various cases of η_escα as labeled. Secondary electrons are produced from proton-induced γ rays absorbed by isotropic external photons. The thick dash-dotted line is the analytical solution of Eq. (14), assuming only one isotropic photon field with n_i = 1 × 10¹⁰ cm⁻³, and z_i = 1 pc. The dashed lines are the standard solution, Eq. (7). The vertical red line is t_esc(0), while the horizontal dotted lines mark the maximum level of the standard solution and when they are reached by the dash-dotted lines. Further parameters are p = 2, q_0, e = q_0, p = 1 cm⁻³ s⁻¹, R₀ = 5 × 10¹⁵ cm, η_esc = 3, and Γ = 10. The solid lines mark simulations with OneHaLe.