HII Regions and Photoionization

An HII region — read "H-two", the shorthand for ionized hydrogen H $^+$ — is the glowing bubble of gas that a hot, young, massive star inflates simply by shining. Only stars hotter than about $25{,}000\,\mathrm{K}$ , the O and early-B types, pour out an appreciable flux of photons past the hydrogen Lyman limit at $\lambda=912\,\mathrm{\AA}$ ( $h\nu=13.6\,\mathrm{eV}$ ), and each such photon can tear an electron off a hydrogen atom. Because those stars live only a few million years, an HII region is a real-time beacon of recent star formation — which is why the Orion and Rosette Nebulae are so conspicuous. The whole structure follows from one bookkeeping identity. Photoionization removes electrons; recombination puts them back; in steady state the two rates match everywhere. Integrated over the volume, that balance says the star's ionizing output equals the total recombination rate — and that single equation, written down by Bengt Strömgren in 1939, fixes the size of the ionized zone, the Strömgren sphere. It also predicts the region's most surprising feature: a razor-thin ionization front one photon mean free path wide, dividing a nearly fully ionized interior from cold neutral gas outside. A second balance sets the temperature: the excess energy each photoelectron carries above threshold heats the gas, and forbidden lines of trace ions radiate it away, pinning the electron temperature near $10^4\,\mathrm{K}$ almost regardless of the star. And because that interior is a hundred times overpressured, the sphere does not sit still — it expands, driving feedback into the very cloud that made the star.

Key takeaways

An HII region is a zone of ionized hydrogen around a hot O or early-B star whose photons past the Lyman limit ( $h\nu>13.6\,\mathrm{eV}$ ) strip electrons from hydrogen atoms.
In steady state the star's ionizing photon rate $Q_0$ equals the total recombination rate; that global balance fixes the size of the ionized zone — the Strömgren sphere.
The Strömgren radius scales as $R_{\mathrm{S}}\propto(Q_0/\alpha_B n^2)^{1/3}$ : it grows only as the cube root of the stellar output but shrinks as $n^{-2/3}$ in dense gas.
The edge is a discontinuity, not a gradient: the ionization front is about one photon mean free path thick, some $10^{-4}$ of $R_{\mathrm{S}}$ .
A separate heating–cooling balance holds the gas near $T\approx10^4\,\mathrm{K}$ ; the recombination cascade lights up H $\alpha$ and the radio continuum that make it observable.
Because the hot interior is overpressured, real HII regions expand as $R\propto t^{4/7}$ at $c_{\mathrm{II}}\approx13\,\mathrm{km\,s^{-1}}$ — a key channel of stellar feedback.
Key idea: The structure of an HII region follows from a single balance — in steady state the number of ionizing photons the star emits per second equals the number of recombinations per second to excited states in the ionized gas. Equating them gives the Strömgren radius and a sharp ionization front, while a second balance of photoionization heating against line cooling holds the gas near $10^4\,\mathrm{K}$ .

The core insight

The structure of an HII region follows from a single balance: in steady state the number of ionizing photons emitted per second by the star equals the number of recombinations per second (to excited states) in the ionized gas. Equating these gives the Strömgren radius and a sharp ionization front, with the gas held near $10^4\,\mathrm{K}$ by the balance of photoionization heating against line cooling.

Where this lives today

The photon-balance picture is a working instrument. Extragalactic surveys turn a galaxy's H $\alpha$ (or radio free–free) luminosity into a star-formation rate through the same bookkeeping worked above — the backbone of the Kennicutt–Schmidt relation and of SFR maps from SDSS, MaNGA, and JWST. Radio and submillimetre arrays (VLA, ALMA) exploit the emission measure to find ultracompact HII regions pinpointing massive protostars in their first $10^5\,\mathrm{yr}$ . Integral-field spectrographs (MUSE, JWST/NIRSpec) resolve the forbidden-line ratios of equation (6) to map temperature, density, and metallicity pixel by pixel. And the expansion law drives current work on stellar feedback — whether HII-region pressure, radiation pressure, or supernovae dominate cloud disruption, a central open problem in star formation. Pushed to high redshift, the same photon budget governs cosmic reionization (see the FAQ below).

The physical picture

Picture the star as a fountain spraying $Q_0$ ionizing photons per second into uniform gas. Each photon that finds a neutral atom ionizes it; meanwhile electrons and protons keep recombining, each recombination "using up" the capacity to have absorbed one photon. Equilibrium arrives when supply matches the total demand from recombinations in the whole volume — that volume is the Strömgren sphere. Turn the star up and it grows; compress the gas and it shrinks steeply, since recombinations go as density squared. The first figure is the anatomy that balance produces: a fully ionized core, a knife-edge front, neutral gas beyond.

Figure

Cross-section of a Strömgren sphere around an O star: ionizing photons Q₀ keep a central H⁺ zone almost fully ionized, are consumed at a razor-thin ionization front, and leave neutral H⁰ gas outside. The balance Q₀ = (4/3)πR_S³ α_B n² gives R_S ≈ 14.6 pc for Q₀ = 10⁴⁹ s⁻¹ and n = 10 cm⁻³.

Why is the edge sharp? The threshold cross section is enormous, $\sigma_{912}\approx6.3\times10^{-18}\,\mathrm{cm}^2$ , so once a photon meets neutral atoms it is absorbed almost at once. Deep inside, only a trace neutral fraction — of order $10^{-4}$ — survives and photons stream freely; but as they are consumed the neutral density climbs, absorption runs away, and the last photons are devoured over a shell one mean free path thick. The result is the near step-function plotted next — a front a thousand times thinner than the sphere.

Figure

The ionization profile is a cliff, not a ramp: the ionized fraction x = n_p / n holds within 10⁻⁴ of unity across the interior, then plunges to zero at R_S ≈ 14.6 pc over a front ℓ ≈ 1/(n_HI σ₉₁₂) ≈ 0.005 pc — only about 3.5×10⁻⁴ of R_S, effectively a discontinuity.

The temperature is set by a different accountant. Each photoionization deposits a few eV (the excess above $13.6\,\mathrm{eV}$ ) as electron kinetic energy, and that heat is bled off by collisionally exciting metastable levels of trace heavy ions, which decay through forbidden lines. Because those coolants have excitation energies of order $k_BT$ at $10^4\,\mathrm{K}$ , the cooling switches on hard right there and the thermostat locks — which is why nebulae glow green in $[\mathrm{O\,III}]$ and red in H $\alpha$ . Finally, the static sphere is only the opening scene: the overpressured interior pushes outward like a balloon, driving a shock and sweeping up a shell — the Strömgren sphere's physics becomes the physics of early feedback.

The core equations

Everything begins with the ionizing budget. Only photons past the Lyman limit can ionize ground-state hydrogen, so what matters is their rate — the specific luminosity integrated over the Lyman continuum, each $L_\nu\,\mathrm{d}\nu$ divided by $h\nu$ .

Q_0=\int_{\nu_1}^{\infty}\dfrac{L_\nu}{h\nu}\,\mathrm{d}\nu \qquad (1)

where: $Q_0$ — number of hydrogen-ionizing (Lyman-continuum) photons emitted per second; $L_\nu$ — stellar specific luminosity (energy per time per frequency); $\nu_1$ — Lyman-limit frequency, $h\nu_1=13.6\,\mathrm{eV}$ , $\lambda_1=912\,\mathrm{\AA}$ ; $h$ — Planck constant. Reading it: the Lyman continuum sits far out on the Wien tail, so $Q_0$ is a violent function of temperature — $\sim10^{49}\,\mathrm{s^{-1}}$ for an O5 star, $\sim10^{46}$ for a B0. The global steady state equates supply to demand. Every ionizing photon is absorbed inside the cloud (ionization-bounded), and each ionization is matched by a recombination, so the emission rate equals the total recombination rate.

\dfrac{4}{3}\pi R_{\mathrm{S}}^3\,\alpha_B\,n_e n_p = Q_0 \qquad (2)

where: $R_{\mathrm{S}}$ — Strömgren radius; $\alpha_B$ — case-B recombination coefficient (to levels $n\ge2$ ), $\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ at $10^4\,\mathrm{K}$ ; $n_e,\,n_p$ — electron and proton densities, with $n_e\approx n_p\approx n$ in pure ionized hydrogen. Reading it: the left side is volume × recombination rate per volume; setting it equal to $Q_0$ is the Strömgren argument. Solving the balance for the radius gives the most-used formula in nebular astrophysics. Every scaling below follows from it.

R_{\mathrm{S}}=\left(\dfrac{3\,Q_0}{4\pi\,\alpha_B\,n^2}\right)^{1/3} \qquad (3)

where: $R_{\mathrm{S}}$ — radius of the ionization-bounded sphere; $n$ — hydrogen number density. Reading it: the cube root makes $R_{\mathrm{S}}$ almost indifferent to the star ( $\propto Q_0^{1/3}$ ) but sharply sensitive to environment ( $\propto n^{-2/3}$ ). The sharp edge comes from radiative transfer. Ionizing photons are removed at a rate set by the neutral density and cross section, so the flux falls exponentially in the neutral column, and locally that flux is exactly what ionization equilibrium demands.

\dfrac{\mathrm{d}F_\nu}{\mathrm{d}r}=-n_{\mathrm{HI}}\,\sigma_\nu\,F_\nu, \qquad n_{\mathrm{HI}}\!\int_{\nu_1}^{\infty}\!\sigma_\nu\dfrac{F_\nu}{h\nu}\,\mathrm{d}\nu=\alpha_A\,n_e n_p \qquad (4)

where: $F_\nu$ — ionizing photon flux; $n_{\mathrm{HI}}$ — neutral hydrogen density; $\sigma_\nu$ — photoionization cross section, $\sigma_{912}\approx6.3\times10^{-18}\,\mathrm{cm^2}$ at threshold; $\alpha_A$ — total (case-A) recombination coefficient. Reading it: the left gives an attenuation length $1/(n_{\mathrm{HI}}\sigma)$ ; the right pins the neutral fraction to the local flux, so where the flux falls $n_{\mathrm{HI}}$ rises and absorption runs away. What we observe is not the ionization but the recombinations. Their brightness is set by the emission measure — the density-squared path integral scaling both the recombination-line and radio free–free glow.

\mathrm{EM}=\int n_e^2\,\mathrm{d}\ell \qquad (5)

where: $\mathrm{EM}$ — emission measure (in $\mathrm{cm^{-6}\,pc}$ ); $n_e$ — electron density; $\mathrm{d}\ell$ — path length along the line of sight. Reading it: because it goes as $n_e^2$ , dense knots outshine diffuse envelopes, and the same integral sets both H $\alpha$ brightness and the radio continuum that reveals ultracompact HII regions. The temperature is fixed by a separate heating–cooling balance. Photoionization heating deposits the photoelectron's excess energy, forbidden lines remove it, and their equality pins $T$ near $10^4\,\mathrm{K}$ .

\Gamma_{\mathrm{pi}}=n_{\mathrm{HI}}\!\int_{\nu_1}^{\infty}\!\dfrac{4\pi J_\nu}{h\nu}\,\sigma_\nu\,(h\nu-h\nu_1)\,\mathrm{d}\nu \;=\; \Lambda_{\mathrm{line}}=n_e\,n_X\,q_{lu}\,\Delta E_{lu}\,\dfrac{A_{ul}}{A_{ul}+n_e q_{ul}} \qquad (6)

where: $\Gamma_{\mathrm{pi}}$ — heating rate per volume; $J_\nu$ — mean intensity; $h\nu-h\nu_1$ — photoelectron energy (a few eV); $\Lambda_{\mathrm{line}}$ — line cooling per volume; $n_X$ — coolant-ion density (e.g. O $^{++}$ ); $q_{lu},q_{ul}$ — collisional excitation/de-excitation rates; $\Delta E_{lu}$ — line energy; $A_{ul}$ — Einstein- $A$ of the decay. Reading it: the factor $A_{ul}/(A_{ul}+n_eq_{ul})$ is the escape probability — it drops above the critical density, which makes forbidden-line ratios density diagnostics. Finally, the region does not stay static. Its $10^4\,\mathrm{K}$ interior has a sound speed of order $10\,\mathrm{km\,s^{-1}}$ , and the pressure imbalance drives it outward on a self-similar track.

c_{\mathrm{II}}=\sqrt{\dfrac{2k_BT_{\mathrm{II}}}{m_{\mathrm{H}}}}\approx13\,\mathrm{km\,s^{-1}}, \qquad R(t)=R_{\mathrm{S}}\left(1+\dfrac{7}{4}\dfrac{c_{\mathrm{II}}t}{R_{\mathrm{S}}}\right)^{4/7} \qquad (7)

where: $c_{\mathrm{II}}$ — isothermal sound speed of the ionized gas (the factor 2 because ionization doubles the particle count); $T_{\mathrm{II}}\approx10^4\,\mathrm{K}$ ; $m_{\mathrm{H}}$ — hydrogen mass; $R(t)$ — Spitzer expansion radius. Reading it: at early times $R\approx R_{\mathrm{S}}$ , at late times $R\propto t^{4/7}$ , and the front decelerates as the shell gains mass.

Derivation AThe Strömgren radius from a global photon budget

The goal is to derive the size of a static HII region from conservation of photons alone. We place one star of output $Q_0$ in uniform, pure-hydrogen gas of density $n$ , assume steady state so ionizations and recombinations balance, and assume it is ionization-bounded — every photon is absorbed inside rather than leaking out. From these three assumptions the radius follows in four lines, and the cube-root scaling that dominates the subject drops out.

Step 1. In an ionization-bounded region every photon that leaves the star is absorbed, so the emission rate equals the total ionization rate over the volume: $Q_0=\int_V (\text{ionization rate per volume})\,\mathrm{d}V$ . Why: this "no photons escape" statement converts a property of the star into one of the gas volume.

Step 2. In steady state each ionization is matched by a recombination, so the ionization rate per volume equals $\alpha_B\,n_e n_p$ , giving $Q_0=\int_V \alpha_B\,n_e n_p\,\mathrm{d}V$ . Why: we use $\alpha_B$ , not $\alpha_A$ — a ground-state recombination emits a Lyman-continuum photon reabsorbed on the spot (the on-the-spot approximation).

Step 3. For a uniform sphere with $n_e\approx n_p\approx n$ out to $R_{\mathrm{S}}$ , the integral is trivial: $Q_0=\alpha_B\,n^2\cdot\tfrac{4}{3}\pi R_{\mathrm{S}}^3$ . Why: constant density pulls $\alpha_B n^2$ out, leaving only the geometric volume.

Step 4. Solve: $R_{\mathrm{S}}=\left(3Q_0/4\pi\alpha_B n^2\right)^{1/3}$ . Substituting $Q_0=10^{49}\,\mathrm{s^{-1}}$ , $n=10\,\mathrm{cm^{-3}}$ , $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ gives $R_{\mathrm{S}}\approx4.5\times10^{19}\,\mathrm{cm}\approx14.6\,\mathrm{pc}$ . Why: the cube root makes the result robust — a large error in $Q_0$ moves $R_{\mathrm{S}}$ by only a factor of two. *As a check: $(3\times10^{49})/(4\pi\cdot2.6\times10^{-13}\cdot10^2)=9.2\times10^{58}\,\mathrm{cm^3}$ , whose cube root is $4.5\times10^{19}\,\mathrm{cm}=14.6\,\mathrm{pc}$ ; the $n^{-2/3}$ scaling correctly recovers $9.8\times10^{18}\,\mathrm{cm}$ at $n=100\,\mathrm{cm^{-3}}$ .*

Go deeper · Derivation B · 1 minWhy the ionization front is razor thin

Derivation BWhy the ionization front is razor thin

The goal is to show quantitatively that the ionized-to-neutral transition is not a gradient but a shell only one photon mean free path thick, negligible against $R_{\mathrm{S}}$ . The argument couples two facts already in hand: the neutral fraction is fixed locally by ionization equilibrium against the surviving flux, and that flux is attenuated exponentially over the neutral column. Tracking the flux outward, the neutral density stays tiny until, near the edge, it climbs to the ambient value and consumes the last photons.

Step 1. Local balance reads $n_{\mathrm{HI}}\,\phi\,\sigma\approx\alpha_B n^2$ , where $\phi=\int(F_\nu/h\nu)\,\mathrm{d}\nu$ is the photon flux, so $n_{\mathrm{HI}}/n\approx\alpha_B n/(\phi\sigma)$ . Why: the neutral density is inversely proportional to the local flux, so the deep interior is ionized to a part in $10^4$ .

Step 2. The flux is attenuated with optical depth $\tau=\int n_{\mathrm{HI}}\sigma\,\mathrm{d}r$ , dropping by $e^{-1}$ over the column where $\tau=1$ , i.e. over $\Delta r\sim 1/(n_{\mathrm{HI}}\sigma)$ . Why: this is the photon mean free path — the only length over which the ionization state can change appreciably.

Step 3. These feed back: as $\phi$ falls near the edge, Step 1 forces $n_{\mathrm{HI}}$ up, which by Step 2 shortens the mean free path, driving $\phi$ down still faster. The runaway consumes the last photons over a shell where $n_{\mathrm{HI}}\to n$ , of thickness $\ell\sim1/(n\sigma_{912})$ . Why: the positive feedback turns a smooth exponential into a near-discontinuity.

Step 4. In numbers, with $n=10\,\mathrm{cm^{-3}}$ and $\sigma_{912}=6.3\times10^{-18}\,\mathrm{cm^2}$ , $\ell\approx1.6\times10^{16}\,\mathrm{cm}\approx0.005\,\mathrm{pc}$ , about $3.5\times10^{-4}$ of $R_{\mathrm{S}}=14.6\,\mathrm{pc}$ . Why: a few parts in ten thousand is, for any imaging purpose, a hard edge — hence the crisp rims of HII regions and the thin photodissociation skin outside. *As a check: $\ell=1/(10\cdot6.3\times10^{-18})=1.6\times10^{16}\,\mathrm{cm}$ , and $\ell/R_{\mathrm{S}}=1.6\times10^{16}/4.5\times10^{19}=3.5\times10^{-4}$ .*

Go deeper · Derivation C · 1 minPressure-driven expansion and the Spitzer solution

Derivation CPressure-driven expansion and the Spitzer solution

The goal is to see why a real HII region cannot remain static and to obtain the law by which it grows. The engine is the pressure contrast: the $\sim10^4\,\mathrm{K}$ interior vastly outweighs the cold surroundings, driving a shock that sweeps ambient gas into a dense shell. Balancing the interior pressure against the shell's momentum yields a self-similar equation of motion whose exponent, $4/7$ , marks the class of momentum-driven expansions.

Step 1. Compare the pressures. Inside, $P_{\mathrm{II}}=2n_{\mathrm{II}}k_BT_{\mathrm{II}}$ (the factor 2 counts electrons plus ions), $T_{\mathrm{II}}\sim10^4\,\mathrm{K}$ ; outside, $P_0=n_0k_BT_0$ , $T_0\sim100\,\mathrm{K}$ . The ratio is of order $10^2$ . Why: a hundredfold overpressure cannot be held back — the equilibrium sphere is only an initial condition.

Step 2. The overpressure drives a shock ahead of the front and piles swept-up gas into a thin dense shell; equating the interior pressure to the change of the shell's momentum gives the equation of motion (Spitzer; Dyson & Williams). Why: radiative losses keep the shell thin, so momentum (not energy) is conserved — this fixes the exponent.

Step 3. The similarity solution is $R(t)=R_{\mathrm{S}}\left(1+\tfrac{7}{4}c_{\mathrm{II}}t/R_{\mathrm{S}}\right)^{4/7}$ with $c_{\mathrm{II}}=\sqrt{2k_BT_{\mathrm{II}}/m_{\mathrm{H}}}\approx13\,\mathrm{km\,s^{-1}}$ . Early on $R\approx R_{\mathrm{S}}$ ; late on $R\propto t^{4/7}$ and $v$ falls below $c_{\mathrm{II}}$ . Why: the front decelerates as each parsec loads more mass onto the shell.

Step 4. For our $R_{\mathrm{S}}=14.6\,\mathrm{pc}$ region the shell reaches $\approx25\,\mathrm{pc}$ at $1\,\mathrm{Myr}$ ( $\approx8.6\,\mathrm{km\,s^{-1}}$ ) and $\approx40\,\mathrm{pc}$ by $3\,\mathrm{Myr}$ . Why: over the O star's few-Myr life this disperses much of the parent cloud — a leading channel of pre-supernova feedback that regulates star-formation efficiency. *As a check: the $4/7$ exponent follows from the momentum-driven analysis; $c_{\mathrm{II}}=\sqrt{2\cdot1.38\times10^{-16}\cdot10^4/1.67\times10^{-24}}=12.9\,\mathrm{km\,s^{-1}}$ ; and $R(1\,\mathrm{Myr})=25\,\mathrm{pc}$ .*

Limiting cases

Case	Result	Meaning
Uniform density, single star	$R_{\mathrm{S}}=\left(\dfrac{3Q_0}{4\pi\alpha_B n^2}\right)^{1/3}$	The classical Strömgren radius; the whole model reduces to one photon-balance equation.
Density scaling	$R_{\mathrm{S}}\propto n^{-2/3},\ t_{\mathrm{rec}}=(\alpha_B n)^{-1}$	Denser gas gives smaller, faster-recombining regions; emission measure and radio brightness rise steeply.
Ionization- vs density-bounded	$\int_0^{R}4\pi r^2\alpha_B n_e n_p\,\mathrm{d}r\ \lessgtr\ Q_0$	If the cloud absorbs all photons it is ionization-bounded (a true Strömgren sphere); if photons escape it is density-bounded and leaks radiation.
Sharp ionization front	$\ell=\dfrac{1}{n_{\mathrm{HI}}\sigma_{912}}\ll R_{\mathrm{S}}$	One mean free path wide, $\sim3.5\times10^{-4}R_{\mathrm{S}}$ — effectively a discontinuity.
Dynamical (D-type) expansion	$R(t)=R_{\mathrm{S}}\left(1+\tfrac{7}{4}\dfrac{c_{\mathrm{II}}t}{R_{\mathrm{S}}}\right)^{4/7}$	Once pressure-driven, the region expands at $\sim c_{\mathrm{II}}\approx13\,\mathrm{km\,s^{-1}}$ , decelerating as the shell grows.

Common trapthe region is not held up by α_A, and the edge is not a gradient

Two errors recur. First, using the total coefficient $\alpha_A$ (recombinations to all levels) instead of $\alpha_B$ (to $n\ge2$ ) in the balance. In an optically thick nebula a recombination straight to the ground state emits a Lyman-continuum photon reabsorbed almost immediately, so it never drains the budget — counting it double-counts. The on-the-spot approximation is exactly "use $\alpha_B$ ", and it is why the sphere is a few percent larger than a naive $\alpha_A$ estimate. Second, imagining the front as a slow fade. It is not: because $\sigma_{912}$ is enormous, the neutral fraction leaps from $\sim10^{-4}$ to near unity over one mean free path, a thousand times thinner than $R_{\mathrm{S}}$ . Treating it as a ramp misplaces the 21 cm emission and the photodissociation skin that begin abruptly just outside.

Worked examples

1 · Size, mass, and glow of an O-star HII region (fully worked)

An O5 star with $Q_0=10^{49}\,\mathrm{s^{-1}}$ sits in uniform hydrogen of $n=10\,\mathrm{cm^{-3}}$ at $T=10^4\,\mathrm{K}$ , where $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ . Find the Strömgren radius, the recombination time, the ionized mass, and the H $\alpha$ luminosity.

Step 1. Strömgren radius from equation (3): $R_{\mathrm{S}}=\left(3\times10^{49}/(4\pi\cdot2.6\times10^{-13}\cdot10^2)\right)^{1/3}$ . The ratio inside is $9.2\times10^{58}\,\mathrm{cm^3}$ , whose cube root is $R_{\mathrm{S}}\approx4.5\times10^{19}\,\mathrm{cm}\approx14.6\,\mathrm{pc}$ .

Step 2. Recombination time $t_{\mathrm{rec}}=(\alpha_B n)^{-1}=3.8\times10^{11}\,\mathrm{s}\approx\mathbf{1.2\times10^4\,yr}$ — far shorter than the star's life, so steady state is an excellent approximation.

Step 3. Ionized mass is the volume times $n\,m_{\mathrm{H}}$ : $M=\tfrac{4}{3}\pi R_{\mathrm{S}}^3\,n\,m_{\mathrm{H}}=6.4\times10^{36}\,\mathrm{g}\approx\mathbf{3200\,M_\odot}$ — a single O star ionizes a few thousand solar masses.

Step 4. H $\alpha$ luminosity. In case B about $0.45$ recombinations yield an H $\alpha$ photon, and the recombination rate equals $Q_0$ , so the H $\alpha$ photon rate is $0.45\,Q_0=4.5\times10^{48}\,\mathrm{s^{-1}}$ . Each photon carries $hc/6563\,\mathrm{\AA}=3.0\times10^{-12}\,\mathrm{erg}$ , hence $L_{\mathrm{H}\alpha}\approx\mathbf{1.4\times10^{37}\,erg\,s^{-1}}\ (\approx3600\,L_\odot)$ . Through Kennicutt's $\mathrm{SFR}\simeq7.9\times10^{-42}L_{\mathrm{H}\alpha}$ this maps to $\sim10^{-4}\,M_\odot\,\mathrm{yr^{-1}}$ — a galaxy's H $\alpha$ output literally counts its O stars.

2 · Sharpness of the front and the pace of expansion

For the same region, how thick is the ionization front compared with $R_{\mathrm{S}}$ , and how large has the region grown after $1\,\mathrm{Myr}$ of pressure-driven expansion?

Step 1. Front thickness from one mean free path with $n_{\mathrm{HI}}\to n=10\,\mathrm{cm^{-3}}$ : $\ell=1/(n\sigma_{912})=\mathbf{1.6\times10^{16}\,cm\approx0.005\,pc}$ . Relative to $R_{\mathrm{S}}=14.6\,\mathrm{pc}$ that is $\ell/R_{\mathrm{S}}\approx3.5\times10^{-4}$ — a knife edge.

Step 2. Ionized sound speed from equation (7): $c_{\mathrm{II}}=\sqrt{2k_BT/m_{\mathrm{H}}}=\mathbf{13\,km\,s^{-1}}$ , comfortably supersonic in the $\sim1\,\mathrm{km\,s^{-1}}$ neutral medium, so a shock is driven.

Step 3. Expansion after $t=1\,\mathrm{Myr}=3.16\times10^{13}\,\mathrm{s}$ : the argument $\tfrac{7}{4}c_{\mathrm{II}}t/R_{\mathrm{S}}=1.6$ , so $R=R_{\mathrm{S}}(1+1.6)^{4/7}=14.6\cdot(2.6)^{0.571}\approx\mathbf{25\,pc}$ , at velocity $v=c_{\mathrm{II}}(2.6)^{-3/7}\approx\mathbf{8.6\,km\,s^{-1}}$ . In one million years the sphere has nearly doubled and is still driving a shell into the cloud.

Physics of the portrait

Behind the whole subject stands a single 1939 paper: Strömgren realized that the size of an ionized nebula need not be computed atom by atom — a global photon count would do — fixing the language nebular astronomers still use.

Origins · 1 minBengt Strömgren and how this was discovered

Bengt Strömgren

1908–1987 · nebular astrophysics and the ionization of the interstellar gas

Bengt Georg Daniel Strömgren was a Danish astrophysicist, son of the astronomer Elis Strömgren, who came of age just as quantum mechanics was making stellar and nebular spectra computable. In a 1939 paper in the Astrophysical Journal he asked a deceptively simple question: how large is the region of hydrogen that a hot star can keep ionized? Rather than solving the ionization state everywhere, he argued globally — in equilibrium the star's Lyman-continuum output must equal the total recombination rate in the surrounding gas — and from that single balance obtained both the sharp division of interstellar hydrogen into ionized and neutral zones and the radius of the sphere that now bears his name. He went on to pioneer quantitative stellar spectroscopy and the four-colour uvby photometric system, and directed major institutions on both sides of the Atlantic before returning to Copenhagen. His taste for the decisive simplifying argument, married to an observer's grasp of real spectra, is exactly the style the Strömgren sphere embodies.

History

1864

— Huggins finds bright emission lines in a nebular spectrum, proving the gas is not unresolved stars

1927

— Bowen identifies the green nebular lines as forbidden transitions of doubly ionized oxygen

1939

— Strömgren derives the radius of the ionized-hydrogen zone around a hot star

1954

— Kahn analyzes the pressure-driven expansion and classifies R-type and D-type ionization fronts

1989

— Osterbrock's Astrophysics of Gaseous Nebulae systematizes the temperature and density diagnostics

Going further

Texts. Osterbrock & Ferland, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, 2nd ed., Chs. 2–5 (the definitive treatment) · Draine, Physics of the Interstellar and Intergalactic Medium, Chs. 15–16, 27–37 · Spitzer, Physical Processes in the Interstellar Medium, Chs. 5–6, 12 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §12.2 · Tielens, The Physics and Chemistry of the ISM, Chs. 7–9. The key numerical benchmarks are: (1) $R_{\mathrm{S}}=(3Q_0/4\pi\alpha_B n^2)^{1/3}=4.5\times10^{19}\,\mathrm{cm}\approx14.6\,\mathrm{pc}$ for $Q_0=10^{49}\,\mathrm{s^{-1}}$ , $n=10\,\mathrm{cm^{-3}}$ , $\alpha_B=2.6\times10^{-13}$ (the $n^{-2/3}$ scaling gives $9.8\times10^{18}\,\mathrm{cm}$ at $n=100\,\mathrm{cm^{-3}}$ ); (2) $\ell=1/(n\sigma_{912})=1.6\times10^{16}\,\mathrm{cm}$ , $\sigma_{912}=6.3\times10^{-18}\,\mathrm{cm^2}$ ; (3) $c_{\mathrm{II}}=\sqrt{2k_BT/m_{\mathrm{H}}}\approx13\,\mathrm{km\,s^{-1}}$ ; (4) ionized mass $\approx3200\,M_\odot$ , $L_{\mathrm{H}\alpha}\approx1.4\times10^{37}\,\mathrm{erg\,s^{-1}}$ ; (5) the Spitzer exponent $4/7$ , $R(1\,\mathrm{Myr})\approx25\,\mathrm{pc}$ .

Prerequisites. Phases of the Interstellar Medium · Interstellar Gas Heating and Cooling Processes · Interstellar Dust Grains. Leads to. Molecular Clouds and Giant Molecular Clouds · The 21 cm Line of Neutral Hydrogen · Thermal Dust Emission and Infrared Radiation.

Ask the tutor: Ask me to walk through the Strömgren balance for any star and density, to show why the ionization front is a knife edge, or to connect a nebula's H $\alpha$ luminosity to the number of O stars lighting it.

Why does the Strömgren radius grow so slowly with the star's brightness?

Because it is a volume that must keep up with the photons, not a radius. The total recombination rate scales as the ionized volume, $\propto R^3$ , and in steady state that equals $Q_0$ , so $R\propto Q_0^{1/3}$ . A star ten times more prolific therefore enlarges the sphere by only $10^{1/3}\approx2.15$ . The cube root also makes the model forgiving — a factor-of-a-few error in $Q_0$ barely moves the size — which is why Strömgren's three-line estimate holds up against elaborate codes.

What is the difference between case A and case B recombination?

Case A counts recombinations to all levels, including directly to the ground state ( $n=1$ ); case B counts only those to excited levels ( $n\ge2$ ). The distinction matters because a ground-state recombination emits a photon just above the Lyman limit, able to ionize another atom. In an optically thick nebula that photon is reabsorbed almost immediately, producing no net loss, so the cleanest bookkeeping ignores both it and the recombination that made it — the "on-the-spot" approximation, exactly what using $\alpha_B$ rather than $\alpha_A$ accomplishes.

Why is the electron temperature almost independent of the star?

Because temperature is decided by a heating–cooling balance, not by how many ionizations happen. Each photoionization deposits only a few eV (the excess above $13.6\,\mathrm{eV}$ ) as electron kinetic energy, and this heat is removed by collisionally exciting metastable levels of trace ions (O $^{+}$ , O $^{++}$ , N $^{+}$ ) that radiate forbidden lines. Those levels have excitation energies of order $k_BT$ at $10^4\,\mathrm{K}$ , so cooling rises steeply right there and acts as a thermostat. Turning up $Q_0$ makes the region bigger but does not change the per-particle balance, so $T$ stays pinned near $10^4\,\mathrm{K}$ .

How can astronomers measure the density and temperature of a nebula remotely?

Through forbidden-line ratios, from the cooling physics of equation (6). Two lines from one ion with different critical densities — the $[\mathrm{S\,II}]$ doublet at 6716 and 6731 $\mathrm{\AA}$ — respond differently to collisional de-excitation, so their ratio measures $n_e$ . Two lines from levels of very different excitation energy — $[\mathrm{O\,III}]$ 4363 versus 4959 plus 5007 $\mathrm{\AA}$ — have a ratio set by the Boltzmann factor, so it measures $T_e$ . Being ratios within one ion, they are largely free of abundance and reddening uncertainties — the workhorse nebular diagnostics.

Is the Strömgren sphere ever really a sphere?

Rarely in detail, but the balance still governs the size. The derivation assumes a single star in uniform gas, whereas real clouds are clumpy with the ionizing cluster often near an edge. When photons escape through a low-density channel the region becomes density-bounded there and leaks radiation — the champagne flows and blister shapes seen in Orion. Even so, the global budget $Q_0=\int\alpha_B n_e n_p\,\mathrm{d}V$ still fixes the total ionized volume; only its shape changes, and three-dimensional codes conserve exactly the quantity Strömgren balanced.

What does an HII region have to do with cosmic reionization?

Everything, scaled up. The same balance that carves a $14.6\,\mathrm{pc}$ bubble around one O star governs the ionized bubbles the first galaxies blew in the neutral intergalactic medium a few hundred million years after the Big Bang. Each source ionizes a Strömgren-like zone; as sources multiply the zones overlap until intergalactic hydrogen is fully ionized — the epoch of reionization. The difference is that the IGM recombination time is comparable to the age of the Universe, but the governing budget is the direct descendant of Strömgren's 1939 equation.