Strömgren Spheres

Take a hot, massive star and drop it into a cloud of cold neutral hydrogen. Above $13.6\,\mathrm{eV}$ (wavelengths shorter than $912\,\text{\AA}$ ) its photons tear the electron off a hydrogen atom, and a young O star pours out these Lyman-continuum photons at a staggering rate — $Q_0\sim10^{49}$ every second. But the gas does not ionize without limit: ionized hydrogen keeps recapturing electrons. Bengt Strömgren showed in 1939 that the two processes settle into a global balance: the ionized volume grows until the total number of recombinations inside it exactly consumes the star's photon output. The result is a nearly fully ionized region — an HII region — with a startlingly sharp edge: the Strömgren sphere. The whole model rests on one line of bookkeeping. In steady state the star cannot ionize gas faster than it recombines, so $Q_0$ must equal the volume integral of the recombination rate $\alpha_B\,n_e n_p$ , giving a radius $R_S\propto Q_0^{1/3}n^{-2/3}$ — a few parsecs to a few tens of parsecs for real stars and densities. Two subtleties make the argument sing. First, only recombinations to excited levels count: a capture straight to the ground state emits a fresh ionizing photon, instantly reabsorbed, so it cancels itself — the "on-the-spot" approximation, which replaces the full coefficient $\alpha_A$ with the Case B value $\alpha_B$ . Second, the boundary is sharp by necessity, not assumption: neutral hydrogen is so opaque to ionizing photons that the transition spans a shell a thousand times thinner than $R_S$ . Rescaled, the same photon-budget argument describes ionized bubbles growing in the intergalactic medium during cosmic reionization — Strömgren's little sphere is the template for how starlight carves the universe.

Key takeaways

A Strömgren sphere is the idealized, sharply bounded region of fully photoionized hydrogen around a hot star in uniform neutral gas — the physicist's model of an HII region; its size follows from a photon budget balancing the star's ionizing output ( $h\nu>13.6\,\mathrm{eV}$ ) against total recombinations to excited states.
The classic result $R_S=\left(3Q_0/4\pi n^2\alpha_B\right)^{1/3}$ gives $\approx 68\,\mathrm{pc}$ for an O star ( $Q_0=10^{49}\,\mathrm{s^{-1}}$ ) in diffuse gas ( $n=1\,\mathrm{cm^{-3}}$ ), shrinking to sub-parsec in dense cores because $R_S\propto n^{-2/3}$ .
Only Case B recombinations enter: ground-state captures ( $n=1$ ) emit an ionizing photon reabsorbed on the spot, so they relocate the ionization rather than destroy a photon — use $\alpha_B\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ , not $\alpha_A$ .
The ionized-to-neutral edge is a genuine knife-edge ionization front, thinner than $R_S$ by $\sim10^{-3}$ , because the ionizing-photon mean free path in neutral gas is tiny.
The model underpins HII regions, ultracompact HII regions, ionization-front dynamics, Lyman-continuum escape, and the growth of cosmic-reionization bubbles.
Key idea: in photoionization equilibrium the ionized volume adjusts so that the total number of recombinations to excited states exactly consumes the star's ionizing output: $Q_0=\tfrac{4}{3}\pi R_S^3\,n^2\alpha_B$ .

The core insight

In photoionization equilibrium the ionized volume adjusts so that the total number of recombinations — counting only captures to excited states ( $n\geq2$ ) — exactly consumes the star's ionizing-photon output: $Q_0=\tfrac{4}{3}\pi R_S^3\,n^2\alpha_B$ .

Where this lives today

Strömgren's balance is the working backbone of modern nebular and reionization astrophysics. Radio and infrared surveys catalogue thousands of Milky Way HII regions, and inverting the relation — using a region's size and density to infer the embedded stars' ionizing output $Q_0$ — weighs dust-shrouded O stars and maps where massive stars form. At the dense extreme, ALMA and the VLA resolve ultracompact HII regions hundredths of a parsec across, exactly the sub-parsec spheres equation (3) predicts for $n\sim10^4$ – $10^6\,\mathrm{cm^{-3}}$ , flagging massive stars still swaddled in their birth clouds. The same budget separates ionization-bounded galaxies (a closed Strömgren surface traps every photon) from density-bounded ones (the gas runs out first, so Lyman-continuum radiation leaks out) — the distinction governing whether early galaxies could reionize the universe. And on the largest scale, 21-cm experiments (LOFAR, HERA, the Square Kilometre Array) model the Epoch of Reionization as overlapping ionized bubbles whose growth is Strömgren's balance rewritten with a galaxy's output, the mean cosmic density, and a clumping factor.

The physical picture

Picture the star as a photon factory stamping out $Q_0$ ionizing photons a second, and the surrounding gas as a sponge that soaks them up through recombinations, each excited-level capture "spending" one photon's worth of the budget. In equilibrium the sponge — the ionized volume — grows to exactly the size where its recombination rate equals the supply: no larger, no smaller. Because recombination needs both an electron and a proton, its rate per volume scales as $n^2$ , so the balance is between $Q_0$ and $n^2$ times a volume. That is why the radius scales as $n^{-2/3}$ , not $n^{-1}$ : volume goes as $R_S^3$ . The first figure fixes this picture and pins its number.

Figure

Cross-section of a Strömgren sphere: an O star (Q₀ = 10⁴⁹ s⁻¹) ionizes hydrogen out to a sharp front where the star's photon supply is exactly balanced by recombinations, Q₀ = (4/3)π R_S³ n² α_B, giving R_S ≈ 68 pc in gas of density n = 1 cm⁻³

The boundary is sharp because neutral hydrogen is enormously opaque to its own ionizing photons. The threshold cross section is large, $\sigma_0\approx6.3\times10^{-18}\,\mathrm{cm^2}$ , so a photon straying into even slightly neutral gas is absorbed within a mean free path $\sim(n_{\mathrm{HI}}\sigma_0)^{-1}$ — only $\sim0.05\,\mathrm{pc}$ at $n=1\,\mathrm{cm^{-3}}$ , less in denser gas. The ionization fraction therefore plunges from near unity to near zero across a shell thousands of times thinner than the sphere: where one expects a gentle fade, the physics delivers a wall. The most consequential number is the density scaling: denser gas recombines faster (as $n^2$ ), strangling the sphere — an O star that lights up 68 parsecs of diffuse gas ionizes barely a parsec of a cloud core. The second figure turns that into the working curve an observer reaches for, anchored to real nebulae.

Figure

The working scaling R_S ∝ n^(−2/3) for a fixed O star Q₀ = 10⁴⁹ s⁻¹: the Strömgren radius falls from ≈68 pc in diffuse gas (n = 1) through ≈3.2 pc in a classical HII region (n = 100) to ≈0.15 pc in an ultracompact HII region (n = 10⁴), each factor of 100 in density shrinking R_S by 100^(2/3) ≈ 21.5

The core equations

The chain of reasoning is short: count the photons supplied, count the recombinations demanded, set them equal, solve for the radius. Everything else — the coefficient, the sharp edge, the later expansion — hangs off that one balance. The supply term is the rate at which the star emits ionizing photons: an integral of its spectrum above the Lyman limit, dominated for a hot star by the far ultraviolet.

Q_0 = \int_{\nu_0}^{\infty} \frac{L_\nu}{h\nu}\,\mathrm{d}\nu \qquad (1)

where: $Q_0$ — H-ionizing photons emitted per second ( $\mathrm{s^{-1}}$ ); $L_\nu$ — stellar specific luminosity ( $\mathrm{erg\,s^{-1}\,Hz^{-1}}$ ); $\nu_0$ — threshold frequency, $h\nu_0=13.6\,\mathrm{eV}$ , i.e. $\lambda_0=912\,\text{\AA}$ ; $h$ — Planck's constant. Because $Q_0$ climbs steeply with effective temperature, an O star delivers $\sim10^{49}\,\mathrm{s^{-1}}$ while a mid-B star delivers thousands of times fewer. The demand term is the total recombination rate in the ionized volume, where hydrogen is almost fully ionized ( $n_e\approx n_p\approx n$ ); equilibrium equates the two.

Q_0 = \frac{4}{3}\pi R_S^3\, n_e n_p\, \alpha_B \;=\; \frac{4}{3}\pi R_S^3\, n^2\, \alpha_B \qquad (2)

where: $R_S$ — Strömgren radius; $n_e,\,n_p$ — electron and proton densities, both $\approx n$ inside; $n$ — ambient hydrogen density ( $\mathrm{cm^{-3}}$ ); $\alpha_B$ — Case B coefficient. Read it as a ledger: photons in per second (left) equal recombinations out per second (right), and the sphere sizes itself to match them. Solving (2) for the radius gives the headline result — a cube root, because it is a volume, not a length, that the photons must fill.

R_S = \left(\frac{3\,Q_0}{4\pi\, n^2\, \alpha_B}\right)^{1/3} \qquad (3)

where: $R_S$ — radius of the ionized sphere; $Q_0$ — ionizing-photon rate; $n$ — ambient density; $\alpha_B$ — Case B coefficient. The two exponents carry all the intuition: $R_S\propto Q_0^{1/3}$ (hotter stars, bigger spheres) and $R_S\propto n^{-2/3}$ (denser gas, smaller spheres). Which coefficient enters is the subtle heart of the model: recombinations sort by the final level of the captured electron, and only some cost the budget a net photon.

\alpha_A = \sum_{n\geq 1}\alpha_n,\qquad \alpha_B = \sum_{n\geq 2}\alpha_n = \alpha_A - \alpha_1 \qquad (4)

where: $\alpha_n$ — coefficient to level $n$ ; $\alpha_A$ — total (all levels); $\alpha_B$ — Case B, excluding the ground state; $\alpha_1$ — direct captures to $n=1$ . At $T=10^4\,\mathrm{K}$ , $\alpha_A\approx4.2\times10^{-13}$ and $\alpha_1\approx1.6\times10^{-13}$ , leaving $\alpha_B\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ — the value for equation (3). The sphere is not perfectly ionized to the edge; a tiny neutral fraction survives everywhere, set by the local balance of photoionizations and recombinations.

\frac{n_{\mathrm{HI}}}{n_p} = \frac{n_e\,\alpha_B}{\Gamma},\qquad \Gamma = \int_{\nu_0}^{\infty}\frac{4\pi J_\nu}{h\nu}\,\sigma_\nu\,\mathrm{d}\nu \qquad (5)

where: $n_{\mathrm{HI}}$ — neutral hydrogen density; $n_p$ — proton density; $\Gamma$ — photoionization rate per neutral atom; $J_\nu$ — mean intensity; $\sigma_\nu$ — cross section. Deep inside, the intense field makes $\Gamma$ large and $n_{\mathrm{HI}}/n_p\ll1$ ; near the front the field is absorbed away, $\Gamma$ collapses, and the neutral fraction rockets to unity across a very thin shell. The static sphere is only the opening act: once ionized, the $10^4\,\mathrm{K}$ gas is vastly overpressured against its cold surroundings and expands, driving the front past $R_S$ in the pressure-driven Spitzer phase.

R(t) = R_S\left(1 + \frac{7}{4}\,\frac{c_i\, t}{R_S}\right)^{4/7} \qquad (6)

where: $R(t)$ — front radius during expansion; $c_i\sim10\,\mathrm{km\,s^{-1}}$ — sound speed in the ionized gas; $R_S$ — initial Strömgren radius. The static solution (3) sets the starting scale; (6) governs the parsecs-over-megayears growth of mature HII regions.

Derivation AThe Strömgren radius from photon balance

The goal is to obtain the radius of the ionized region from global photon conservation alone, with every assumption explicit: a single star of ionizing output $Q_0$ in uniform hydrogen of density $n$ , a steady state in which ionizations and recombinations balance, and an interior so highly ionized that electron and proton densities both equal $n$ . We build the recombination demand as a volume integral, set it equal to the supply, solve, and check against a real O star.

Step 1. Assert full interior ionization: deep inside, the radiation field keeps hydrogen almost completely ionized, so $n_e\approx n_p\approx n$ . Why: this lets the recombination rate, which needs one electron and one proton, be written with a single ambient density $n$ instead of two unknown position-dependent densities.

Step 2. Write the recombination rate per volume as $r_{\mathrm{rec}}=\alpha_B\,n_e n_p=\alpha_B\,n^2$ with the Case B coefficient. Why: ground-state captures are reabsorbed on the spot and do not drain the budget, so $\alpha_B$ not $\alpha_A$ enters — justified in the next derivation.

Step 3. Integrate over the sphere: with $n$ uniform, $\dot N_{\mathrm{rec}}=\int_0^{R_S}\alpha_B n^2\,4\pi r^2\,\mathrm{d}r=\tfrac{4}{3}\pi R_S^3\,\alpha_B n^2$ . Why: the total demand is the constant per-volume rate times the sphere's volume — the $r^2$ is pure geometry.

Step 4. Impose steady-state balance, $Q_0=\dot N_{\mathrm{rec}}=\tfrac{4}{3}\pi R_S^3\,\alpha_B n^2$ . Why: in equilibrium every ionizing photon is matched by exactly one net (excited-level) recombination — supply equals demand, the whole physical content of the model.

Step 5. Solve for the radius: $R_S=\left(3Q_0/4\pi n^2\alpha_B\right)^{1/3}$ , with its cube-root dependence on $Q_0$ and $n^{-2/3}$ on density. Why: the cube root appears because it is the ionized volume, scaling as $R_S^3$ , that the balance fixes — the radius carries only a third of that scaling. *As a check: for $Q_0=10^{49}\,\mathrm{s^{-1}}$ , $n=1\,\mathrm{cm^{-3}}$ , $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ , the formula gives $R_S=(3\times10^{49}/(4\pi\times2.6\times10^{-13}))^{1/3}=2.09\times10^{20}\,\mathrm{cm}=67.9\,\mathrm{pc}$ ; at $n=100\,\mathrm{cm^{-3}}$ the same formula yields $9.77\times10^{18}\,\mathrm{cm}$ , exactly $67.9\,\mathrm{pc}$ divided by $100^{2/3}$ .*

Go deeper · Derivation B · 1 minWhy Case B and the on-the-spot approximation

Derivation BWhy Case B and the on-the-spot approximation

The goal is to explain why the correct coefficient is $\alpha_B$ , which omits recombinations directly to the ground state, rather than the total $\alpha_A$ . The argument is radiative transfer of the photons that recombinations themselves emit: we track a ground-state-capture photon, show it is reabsorbed almost immediately because the gas is opaque to it, and conclude such captures merely shuffle the ionization without changing the net budget — so only excited-level captures count.

Step 1. Note that a recombination directly to $n=1$ releases a photon with $h\nu\geq13.6\,\mathrm{eV}$ , itself able to ionize another hydrogen atom. Why: the energy liberated is at least the ionization energy, so the photon is a fresh member of the Lyman continuum — a recycled ionizing photon, not a loss.

Step 2. Estimate its fate: the mean free path $\ell_{\mathrm{mfp}}=(n_{\mathrm{HI}}\sigma_0)^{-1}$ is minuscule compared with $R_S$ for any relevant density. Why: the threshold cross section $\sigma_0\approx6.3\times10^{-18}\,\mathrm{cm^2}$ is huge, so the recycled photon is reabsorbed almost where it was emitted — it never escapes.

Step 3. Conclude that each ground-state recombination is undone by the very photon it emits ionizing a neighbour: the ionization is relocated, not destroyed, with no net change to the budget. Why: this "on-the-spot" cancellation makes ground-state captures invisible to the global balance, so they drop from the tally.

Step 4. Count only recombinations to excited levels ( $n\geq2$ ), giving the net rate $\dot N_{\mathrm{net}}=\alpha_B\,n_e n_p\,V$ with $\alpha_B=\alpha_A-\alpha_1$ . Why: excited-level captures emit photons below the Lyman limit that escape or are degraded, so they are genuine permanent losses of ionizing photons — exactly what must balance $Q_0$ . *As a numerical check: at $T=10^4\,\mathrm{K}$ , $\alpha_A\approx4.2\times10^{-13}$ and $\alpha_1\approx1.6\times10^{-13}$ , giving $\alpha_B\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ ; using $\alpha_A$ instead would undersize $R_S$ by $(\alpha_A/\alpha_B)^{1/3}\approx1.17$ .*

Go deeper · Derivation C · 1 minWhy the ionization front is a knife edge

Derivation CWhy the ionization front is a knife edge

The goal is to show that the sharp-boundary idealization is forced by the physics, by estimating the width of the transition zone between ionized and neutral gas and comparing it with the radius. The reasoning is an optical-depth argument: ionizing photons are absorbed once they cross a neutral column of order the inverse cross section, and converting that column to a physical thickness reveals a shell far thinner than $R_S$ .

Step 1. Find the neutral column for unit optical depth: $N_{\mathrm{HI}}\sigma_0\sim1$ gives $N_{\mathrm{HI}}\sim\sigma_0^{-1}\approx1.6\times10^{17}\,\mathrm{cm^{-2}}$ . Why: a photon survives on average until it has traversed one optical depth, so this column marks where the radiation field is extinguished and the gas turns neutral.

Step 2. Convert to a physical width by dividing by the near-neutral density at the front, $\Delta R\sim N_{\mathrm{HI}}/n\approx1.6\times10^{17}/(n/\mathrm{cm^{-3}})\,\mathrm{cm}$ — about $0.05\,\mathrm{pc}$ for $n=1\,\mathrm{cm^{-3}}$ . Why: the column is a line-of-sight integral of density, so dividing by density recovers the shell's length.

Step 3. Compare with the radius: $\Delta R/R_S\sim(n_{\mathrm{HI}}\sigma_0 R_S)^{-1}\approx0.05\,\mathrm{pc}/68\,\mathrm{pc}\sim7\times10^{-4}$ . Why: the ratio is tiny — interior and exterior are separated by a wall three orders of magnitude thinner than the sphere, a knife edge that denser gas only sharpens. *As a numerical check: at $n=1\,\mathrm{cm^{-3}}$ , $\ell_{\mathrm{mfp}}=(1\times6.3\times10^{-18})^{-1}=1.6\times10^{17}\,\mathrm{cm}=0.051\,\mathrm{pc}$ and $R_S=2.09\times10^{20}\,\mathrm{cm}$ , giving $\Delta R/R_S=7.6\times10^{-4}$ — confirming the sharp-boundary treatment.*

Limiting cases

The model has clean asymptotics, and each one exposes a different piece of its structure.

Case	Result	Meaning
Sharp-boundary limit	$\ell_{\mathrm{mfp}}=1/(n\,\sigma_0)\ll R_S$	When the ionizing-photon mean free path is far smaller than the sphere, the transition zone is negligible and the fully-ionized-sphere idealization holds.
Density dependence	$R_S\propto n^{-2/3}$	Denser gas recombines faster, giving smaller, sharper-edged HII regions; a factor 100 in density shrinks $R_S$ by $\approx21.5$ .
Source dependence	$R_S\propto Q_0^{1/3}$	Hotter, more luminous stars make larger spheres; since $Q_0$ climbs steeply with $T_{\mathrm{eff}}$ , O stars dominate and B stars make only modest nebulae.
Dynamical late time	$R(t)\propto t^{4/7}$	Once pressure drives the expansion, the front grows beyond $R_S$ as $t^{4/7}$ , sweeping up a dense shell — the mature-HII-region phase.
Reionization analogue	$V_{\mathrm{ion}}\propto \dot N_{\mathrm{ion}}/(\bar n_H^2\,\alpha_B\,C)$	The same balance sizes cosmic ionized bubbles, with $Q_0\to$ a galaxy's output, $n\to$ the mean cosmic density, and clumping factor $C=\langle n^2\rangle/\langle n\rangle^2$ .

Common trapUse Case B, not Case A — and remember the radius scales as n to the minus two-thirds

The most common mistake is to plug the total coefficient $\alpha_A$ (which sums over all final levels, including the ground state) into the formula. Recombinations straight to $n=1$ emit a fresh ionizing photon reabsorbed locally, so under the on-the-spot approximation they cause no net loss and must be excluded — use $\alpha_B\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ at $10^4\,\mathrm{K}$ , roughly $0.62\,\alpha_A$ . The second trap is dimensional: recombination needs two particles, so the rate goes as $n^2$ and the balance fixes a volume, giving $R_S\propto n^{-2/3}$ , not $n^{-1}$ . A sphere scaling linearly with density means you counted only one factor of $n$ .

Worked examples

1 · Strömgren radius of an O star, and how density shrinks it (fully worked)

An O6 star emits $Q_0=10^{49}$ ionizing photons per second. Find its Strömgren radius in diffuse gas ( $n=1\,\mathrm{cm^{-3}}$ ), then in a classical HII region ( $n=100\,\mathrm{cm^{-3}}$ ), using $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ .

Step 1. Assemble the cube: $R_S^3=3Q_0/(4\pi n^2\alpha_B)$ . For $n=1$ the denominator is $4\pi\times2.6\times10^{-13}=3.27\times10^{-12}$ , so $R_S^3=3\times10^{49}/3.27\times10^{-12}=9.18\times10^{60}\,\mathrm{cm^3}$ .

Step 2. Take the cube root: $R_S=(9.18\times10^{60})^{1/3}=2.09\times10^{20}\,\mathrm{cm}$ ; dividing by $3.086\times10^{18}\,\mathrm{cm\,pc^{-1}}$ gives $R_S\approx68\,\mathrm{pc}$ — a giant diffuse nebula tens of parsecs across.

Step 3. Rescale to $n=100$ using $R_S\propto n^{-2/3}$ : density up by $100$ means $R_S$ down by $100^{2/3}=21.5$ , so $R_S=68/21.5=$ $3.2\,\mathrm{pc}$ — the parsec scale of a classical HII region like the core of Orion.

Step 4. Check the trend against real objects: $n\sim10$ gives $\approx15\,\mathrm{pc}$ (the sprawling Rosette), $n\sim10^4$ gives $\approx0.15\,\mathrm{pc}$ (an ultracompact HII region in a molecular clump) — one cube-root law spanning three orders of magnitude in size from a single star. R_S = 68 pc at n = 1, falling to 3.2 pc at n = 100 — the $n^{-2/3}$ scaling in action.

2 · How fast does the sphere light up?

Estimate the time for the ionized region to reach equilibrium size after the star switches on, for $n=1$ and $n=100\,\mathrm{cm^{-3}}$ , using the recombination time $t_{\mathrm{rec}}=1/(n\alpha_B)$ .

Step 1. At $n=1$ : $t_{\mathrm{rec}}=1/(1\times2.6\times10^{-13})=3.85\times10^{12}\,\mathrm{s}$ ; with $1\,\mathrm{yr}=3.16\times10^7\,\mathrm{s}$ this is $t_{\mathrm{rec}}\approx1.2\times10^5\,\mathrm{yr}$ to fill a diffuse sphere.

Step 2. At $n=100$ the time scales as $1/n$ , so $t_{\mathrm{rec}}=1.2\times10^5/100\approx$ $1200\,\mathrm{yr}$ — denser gas both shrinks the sphere and lights it up far faster.

Step 3. Interpret via the formation law $R_i(t)=R_S[1-e^{-t/t_{\mathrm{rec}}}]^{1/3}$ : after one $t_{\mathrm{rec}}$ the front has reached $(1-e^{-1})^{1/3}=0.86\,R_S$ , and after a few it is essentially at $R_S$ . Since $t_{\mathrm{rec}}$ is short next to a massive star's few-Myr life, the static sphere is established almost instantly on stellar-evolution timescales, and the slow $t^{4/7}$ expansion then takes over. The sphere fills in a few recombination times — $\sim10^5$ yr (diffuse) to $\sim10^3$ yr (dense) — a blink compared with the star's life.

Physics of the portrait

The Strömgren sphere names both a shape and a way of thinking. Its author spent a career turning stellar and interstellar physics into computable models, and the photon-budget argument bearing his name is the prototype for every "ionizing photons in equal recombinations out" calculation in astrophysics.

Origins · 1 minBengt Strömgren and how this was discovered

Bengt Strömgren

1908–1987, Danish astrophysicist · photoionization of the interstellar medium and quantitative stellar atmospheres

Bengt Georg Daniel Strömgren was a prodigy of Danish astronomy — son of the Copenhagen Observatory director Elis Strömgren, he published his first paper at fourteen and held a doctorate by twenty-one. In 1939 he published the paper that defines this article ("The Physical State of Interstellar Hydrogen", ApJ 89, 526), showing that a hot star surrounds itself with a sharply bounded zone of ionized hydrogen whose radius is set by the balance of ionizations and recombinations. It was a landmark: it explained why HII regions have crisp edges, why their sizes depend on stellar type and density in a definite way, and introduced the photon-budget bookkeeping the subject still uses. His reach went far beyond nebulae — he pioneered quantitative model stellar atmospheres, devised the four-colour (uvby) photometric system still in daily use, and from 1957 held Einstein's former chair at the Institute for Advanced Study. The sphere is his most famous legacy, but it is of a piece with his lifelong project: making the physics of stars and their surroundings something you can actually compute.

History

1864

— Huggins' nebular spectra reveal that some nebulae are glowing gas, not unresolved stars

1927

— Bowen identifies the mysterious nebular lines as forbidden transitions of ionized oxygen and nitrogen

1939

— Strömgren derives the radius of a photoionized hydrogen sphere from ionization–recombination balance

1954

— Kahn and others classify R-type and D-type ionization fronts, opening the dynamical theory

2003

— WMAP's reionization optical depth ties the photon-budget argument to the growth of cosmic ionized bubbles

Going further

Texts. Strömgren, B. 1939, ApJ 89, 526 (the original derivation) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 12 (The Interstellar Medium) · Osterbrock & Ferland, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, 2nd ed., ch. 2 (photoionization equilibrium, Case A/B) · Draine, Physics of the Interstellar and Intergalactic Medium, ch. 15 and ch. 37 (HII regions and ionization-front dynamics) · Spitzer, Physical Processes in the Interstellar Medium (expansion) · Dodelson, Modern Cosmology (reionization photon budget) · Mo, van den Bosch & White, Galaxy Formation and Evolution (reionization bubbles). Key anchor values: $R_S=(3Q_0/4\pi n^2\alpha_B)^{1/3}$ gives $R_S=2.09\times10^{20}\,\mathrm{cm}=67.9\,\mathrm{pc}$ for $Q_0=10^{49}\,\mathrm{s^{-1}}$ , $n=1\,\mathrm{cm^{-3}}$ , $\alpha_B=2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ , consistent with $9.77\times10^{18}\,\mathrm{cm}$ at $n=100$ once rescaled by $100^{2/3}$ ; recombination coefficients at $10^4\,\mathrm{K}$ : $\alpha_A\approx4.2\times10^{-13}$ , $\alpha_1\approx1.6\times10^{-13}$ , $\alpha_B\approx2.6\times10^{-13}\,\mathrm{cm^3\,s^{-1}}$ ; photoionization threshold cross section $\sigma_0\approx6.3\times10^{-18}\,\mathrm{cm^2}$ at $13.6\,\mathrm{eV}$ ( $912\,\text{\AA}$ ); recombination time $t_{\mathrm{rec}}=1/(n\alpha_B)\approx1.2\times10^5\,\mathrm{yr}$ at $n=1\,\mathrm{cm^{-3}}$ ; front width $\Delta R/R_S\sim7\times10^{-4}$ .

Prerequisites. Phases of the Interstellar Medium · Interstellar Gas Heating and Cooling Processes · The 21 cm Line of Neutral Hydrogen. Leads to. Thermal Instability and the Two-Phase Medium · Molecular Clouds and Giant Molecular Clouds · Interstellar Dust Grains.

Ask the tutor: Want me to walk through how the same photon-budget line that sizes an HII region gets rewritten to grow ionized bubbles across the reionizing universe?

Is a real HII region actually a perfect sphere with a knife-edge boundary?

The sphere is an idealization, exact only for a single star in perfectly uniform gas. Real interstellar gas is clumpy, magnetized, and often has density gradients, so real HII regions are lumpy and sometimes "blister" open on one side of a cloud (champagne flows). Two predictions survive: the size set by the photon budget, $R_S\propto Q_0^{1/3}n^{-2/3}$ , and the sharpness of the edge. The boundary really is thin — the ionization fraction drops across a shell $\sim10^{-3}$ of the radius — because that follows from the large photoionization cross section, which no amount of clumping removes.

Why exactly must I use Case B rather than the total recombination coefficient?

Because a recombination directly to the ground state ( $n=1$ ) hands back a photon energetic enough to ionize another hydrogen atom, and that photon is reabsorbed almost instantly, since the gas is extremely opaque at the Lyman limit. So a ground-state capture, its re-emission, and its re-absorption are a wash — one atom recombines, a neighbour ionizes, the ionized total is unchanged, and it costs the budget nothing. Only captures to excited levels ( $n\geq2$ ) emit photons that leave the ionizing tally (as Balmer, Paschen, and two-photon radiation), a permanent loss the star must replace. Summing exactly those gives $\alpha_B$ .

Does the ionized sphere keep growing until the star dies?

It grows in two phases. First the fast ionization phase: when the star switches on, the front races out to the static radius $R_S$ within a few recombination times — $\sim10^5\,\mathrm{yr}$ in diffuse gas, far less if dense. Ionization balance is then satisfied and the sphere would sit still but for pressure. Because the ionized gas is at $10^4\,\mathrm{K}$ against surroundings near $100\,\mathrm{K}$ , it is hugely overpressured and begins a slow, pressure-driven dynamical expansion, $R(t)\propto t^{4/7}$ , sweeping up a dense neutral shell. This phase runs for the star's whole few-million-year life and gives mature HII regions their shells and bubbles.

What sets the 10⁴ K temperature inside, and does it feed back on the size?

The temperature is fixed by its own thermal balance, separate from ionization balance: photoionization deposits the excess photon energy above threshold as heat, and the gas cools mainly through collisionally excited forbidden lines of trace ions like O⁺⁺ and N⁺. That settles near $8000$ – $10^4\,\mathrm{K}$ across a wide range of conditions, so $T$ is roughly constant. It feeds back weakly on the size, because $\alpha_B$ falls with temperature (roughly as $T^{-0.8}$ ): a hotter nebula recombines slightly more slowly and is a touch larger. But that is mild next to the strong $Q_0^{1/3}n^{-2/3}$ scaling, so quoting $\alpha_B$ at $10^4\,\mathrm{K}$ is an excellent approximation.

How does a Strömgren sphere connect to cosmic reionization?

By rewriting the same balance on a cosmological stage. Replace the star's $Q_0$ with an early galaxy's ionizing output and the ambient $n$ with the mean density of the intergalactic medium, and the equation predicts the volume of the ionized bubble that galaxy inflates. Two changes matter: the universe expands, so the mean density drops with time, and the gas is clumpy, so recombinations are enhanced by a clumping factor $C=\langle n^2\rangle/\langle n\rangle^2$ in the denominator. As bubbles from many galaxies grow and overlap, the cosmic neutral fraction falls to zero — reionization complete. The physics is Strömgren's; only the scale has changed.

What is the difference between an ionization-bounded and a density-bounded nebula?

Whether the photons or the gas run out first. An ionization-bounded nebula has enough neutral gas to absorb every ionizing photon: the Strömgren surface closes inside the cloud, the edge is sharp, and essentially no Lyman-continuum radiation escapes. A density-bounded nebula runs out of gas first — the would-be $R_S$ exceeds the cloud — so the front never forms and leftover photons leak into intergalactic space. The distinction is central to reionization: only density-bounded (or channel-riddled) galaxies leak the Lyman-continuum photons that ionize the intergalactic medium. You decide it by comparing $R_S$ with the actual extent of the gas.