= Ionization-front growth from an exponentially brightening source
{title2=$\frac{4\pi}3n_{H,0}X^3=Q_0\tau(e^{t/\tau}-1)$}
In a homogeneous initially neutral medium, an isotropic source with <hydrogen-ionizing photon production rate> $Q(t)=Q_0e^{t/\tau}$ 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.
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