Ionization-front growth from an exponentially brightening source ($\frac{4\pi }{3}{n}_{H,0}{X}^3=Q_0\tau (e^{t/\tau }-1)$)

In a homogeneous initially neutral medium, an isotropic source with hydrogen-ionizing photon production rate $Q(t)=Q_0{e}^{t/\tau }$ ionizes the displayed comoving inventory when every escaping photon produces one primary ionization and recombinations are absent. The physical front radius is $a(t)X$ in an expanding background. Retarded photon propagation prevents any instantaneous front from travelling faster than light. This photon-budget front is not the equilibrium Strömgren radius, which balances emission against recombinations.