= Solution
A star-forming galaxy contains short-lived, massive O and B stars whose hot photospheres dominate the ultraviolet and blue continuum and ionize surrounding gas. Once star formation ceases, these stars disappear quickly and an older, cooler stellar population produces a redder spectrum with stronger stellar absorption features and a prominent 4000-angstrom break.
In star-forming regions, direct stellar continuum is accompanied by nebular free-bound and free-free continuum, hydrogen and helium recombination lines, collisionally excited metal lines, and infrared emission from dust that absorbed shorter-wavelength photons. Supernova remnants and cosmic rays add synchrotron radio emission, while hot shocked gas can emit X-rays.
The <Strömgren sphere> model assumes a steady ionizing source in uniform, static, pure hydrogen of number density $n_H$, with a sharp <ionization front> enclosing fully ionized gas. If $N_i=(4\pi/3)R^3n_H$ is the number of ions, photon conservation gives
$$
\frac{dN_i}{dt}=\dot N_{\rm ion}
-\frac{4\pi}{3}R^3\alpha n_H^2.
$$
The equilibrium <Strömgren radius> and <recombination time> are
$$
R_S^3=\frac{3\dot N_{\rm ion}}{4\pi\alpha n_H^2},
\qquad t_{\rm rec}=\frac1{\alpha n_H}.
$$
Consequently the radius obeys
$$
3R^2\frac{dR}{dt}
=\frac{R_S^3-R^3}{t_{\rm rec}}.
$$
Writing $x=(R/R_S)^3$ turns this into $dx/d(t/t_{\rm rec})=1-x$. For an initially neutral medium, $x(0)=0$, and the <Ionization-front growth of a Strömgren sphere> is
$$
\boxed{R(t)=R_S\left(1-e^{-t/t_{\rm rec}}\right)^{1/3}.}
$$
The front initially expands rapidly because few ions are recombining and asymptotically approaches $R_S$ as recombinations balance ionizations. This photon-counting solution precedes any pressure-driven hydrodynamic expansion of the H II region.
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