Interstellar Medium
The Strömgren Sphere
The fully-ionized bubble a hot star carves out of hydrogen — where light-eating balances recombination
A Strömgren sphere is the idealized, fully-ionized sphere of hydrogen surrounding a hot star, where the star's ultraviolet photons ionize the surrounding gas out to a radius at which photoionizations exactly balance recombinations. Danish astronomer Bengt Strömgren worked out the theory in 1939, showing that a hot O or B star does not ionize its surroundings gradually — it carves a sharp-edged bubble of H⁺ whose radius follows R_S ∝ (Q/n²)^(1/3), where Q is the star's output of ionizing photons and n is the gas density. Every photon above the 13.6 eV Lyman limit is consumed inside; beyond a wafer-thin ionization front the gas snaps back to neutral H I. This single balance sets the characteristic size of every H II region — from a 0.1-parsec ultracompact bubble around a young B star to a 30-parsec cavity around an O-star cluster.
- Governing relationR_S = (3Q / 4π α_B n²)^(1/3)
- ScalingR_S ∝ (Q/n²)^(1/3)
- Ionization threshold13.6 eV (Lyman limit, λ ≤ 912 Å)
- Case B coefficientα_B ≈ 2.6×10⁻¹³ cm³ s⁻¹ (at 10⁴ K)
- Front thickness~0.01 pc (one photon mean-free-path)
- Theorized byBengt Strömgren, 1939
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Why the Strömgren sphere matters
- It sizes every H II region. The Strömgren radius is the zeroth-order prediction for how big a glowing ionized nebula around a young star should be — the Orion Nebula, the Rosette, the Lagoon all trace back to it.
- It is a stellar luminosity gauge run backwards. Measure a nebula's size or its Hβ / radio free-free luminosity, and the theory hands you Q, the ionizing output — and therefore the spectral type of the embedded star, even when the star is hidden by dust.
- It launches feedback. The overpressured ionized bubble expands supersonically, drives shocks into molecular clouds, and can trigger — or quench — the next generation of star formation.
- It underpins reionization. The same photoionization-versus-recombination bookkeeping, scaled up, governs how the first galaxies ionized the intergalactic medium at redshift z ≈ 6–10.
- It is the cleanest teaching case in ISM physics. One star, uniform gas, one balance equation — a rare closed-form result in an otherwise messy field.
How it works, step by step
The whole idea rests on a competition between two processes happening in the same gas.
- Photoionization. A hot star (T_eff ≳ 25,000 K for an O star) emits a torrent of ultraviolet photons. Any photon with energy above 13.6 eV — wavelength shorter than 912 Å, the Lyman limit — can knock the electron off a hydrogen atom. Call the number the star emits per second Q (also written S₀ or N_Lyc).
- Recombination. Free electrons and protons occasionally meet and recombine into neutral hydrogen. Because it takes a proton and an electron, the recombination rate per unit volume is n_e · n_p · α_B ≈ n² α_B, where α_B is the Case B recombination coefficient and n is the gas number density.
- Global balance. In equilibrium, every ionizing photon the star emits must eventually be canceled by a recombination somewhere inside the ionized volume. Set the star's photon rate equal to the total recombination rate over the sphere: Q = (4/3)π R_S³ · α_B n².
- Solve for the radius. Rearranging gives R_S = (3Q / 4π α_B n²)^(1/3). The cube root is why the sphere is so forgiving: a factor-of-1000 change in Q moves the radius by only a factor of 10.
- The sharp edge. Neutral hydrogen devours Lyman-continuum photons with a huge cross-section (σ ≈ 6×10⁻¹⁸ cm² at threshold). So the moment the photon supply runs low, the last few ionizing photons are absorbed within a single mean-free-path — a shell only ~0.01 pc thick. The gas flips from ~99.99% ionized to ~99.99% neutral across that thin ionization front.
The key equation and its variables
The Strömgren radius comes from equating photon supply and recombination demand:
Q = (4/3) π R_S³ α_B n² ⟹ R_S = (3Q / 4π α_B n²)^(1/3)
| Symbol | Meaning | Typical value / units |
|---|---|---|
| R_S | Strömgren radius (edge of ionized bubble) | 0.1–30 pc |
| Q | Rate of ionizing (Lyman-continuum) photons from the star | 10⁴⁶–10⁴⁹·⁵ s⁻¹ |
| n | Hydrogen number density (n_e ≈ n_p ≈ n) | 1–10⁴ cm⁻³ |
| α_B | Case B recombination coefficient (10⁴ K) | 2.6×10⁻¹³ cm³ s⁻¹ |
| 13.6 eV | Ionization energy of hydrogen (Lyman limit) | λ = 912 Å |
A convenient scaling, evaluated at T = 10⁴ K, is:
R_S ≈ 3.2 pc × (Q / 10⁴⁹ s⁻¹)^(1/3) × (n / 100 cm⁻³)^(−2/3)
The steep inverse dependence on density (R_S ∝ n^−2/3) is why the same star makes a tiny bright bubble in a dense molecular clump and a vast faint cavity in the diffuse medium.
Worked example — an O6 star in the Orion Nebula
The Orion Nebula (M42) is ionized mainly by θ¹ Orionis C, an O6-ish star with Q ≈ 10⁴⁹ ionizing photons per second, embedded in gas of density n ≈ 3000 cm⁻³ in its dense core. Plugging into the scaling:
R_S ≈ 3.2 pc × (10⁴⁹/10⁴⁹)^(1/3) × (3000/100)^(−2/3) ≈ 3.2 pc × 1 × (30)^(−2/3) ≈ 0.33 pc.
That is roughly the ~0.3-pc scale of Orion's bright ionized core — a good match for a first-principles estimate with a single balance equation. The recombination timescale, t_rec = 1/(α_B n) ≈ 1/(2.6×10⁻¹³ × 3000) s ≈ 1.3×10⁹ s ≈ 40 years, tells you how fast the sphere equilibrates: essentially instantly compared with the ~10⁶-year life of the region.
History — Bengt Strömgren, 1939
Bengt Strömgren (1908–1987), a Danish astrophysicist then at the University of Chicago, published "The Physical State of Interstellar Hydrogen" in the Astrophysical Journal in 1939. His insight was that interstellar hydrogen near a hot star is not partially ionized everywhere — it is almost completely ionized inside a well-defined radius and almost completely neutral outside it, with a razor-thin boundary between. He derived the R_S ∝ (Q/n²)^(1/3) relation and computed radii for stars of different temperatures. The idealized bubble has carried his name ever since, and the framework grew into the modern theory of H II regions, ultracompact H II regions, and cosmic reionization.
Common misconceptions
- The star ionizes gas smoothly, fading with distance. No — the transition is a sharp front, not a gradient. Hydrogen's large absorption cross-section makes the edge one photon mean-free-path thick.
- Bigger star ⇒ proportionally bigger sphere. Only as the cube root of Q. Doubling the ionizing output grows the radius by just ~26%.
- Denser gas makes a bigger bubble. The opposite — R_S shrinks as n^−2/3, because recombinations (∝ n²) drain the photon budget faster.
- A Strömgren sphere is a real, static object. It is an idealization. Real ionized bubbles are overpressured and expand supersonically, driving a shock; the observed radius after ~1 Myr exceeds the initial static Strömgren radius.
- It only concerns hydrogen. Hydrogen sets the size, but the same photon-budget logic gives nested He⁺ and He²⁺ Strömgren zones for stars hot enough to ionize helium (24.6 and 54.4 eV).
- Case A recombination is the right coefficient. Use Case B (α_B) — recombinations directly to the ground state emit a photon that is immediately reabsorbed nearby, so they don't count against the global budget.
Frequently asked questions
What is a Strömgren sphere?
It is the idealized, fully-ionized sphere of hydrogen around a hot star, where the star's ultraviolet photons (above 13.6 eV) ionize the surrounding gas out to a radius where photoionizations exactly balance recombinations. Named for Danish astronomer Bengt Strömgren, who worked out the theory in 1939. Inside the sphere hydrogen is essentially all H+; outside it is neutral H I. The boundary between them — the ionization front — is remarkably sharp, only about 0.01 parsec thick in typical conditions.
What sets the radius of a Strömgren sphere?
Global balance: every ionizing photon the star emits per second (the rate Q, sometimes written S or N_Lyc) must be absorbed by a recombination somewhere inside the sphere. Recombinations happen throughout the ionized volume at a rate proportional to n² (electron times proton density), so Q = (4/3)π R_S³ α_B n². Solving gives R_S = (3Q / 4π α_B n²)^(1/3), i.e. R_S ∝ (Q/n²)^(1/3). The α_B is the Case B recombination coefficient, about 2.6×10⁻¹³ cm³/s at 10,000 K.
Why is the ionization front so sharp?
Because neutral hydrogen absorbs Lyman-continuum photons with a large cross-section (~6×10⁻¹⁸ cm² at the Lyman limit). The transition from fully ionized to fully neutral happens over one photon mean-free-path in neutral gas, which is tiny compared with the sphere itself — roughly 0.01 pc versus several pc. So the edge is a thin shell, not a gradual fade. This is why H II regions look like crisp bubbles rather than soft halos.
How big is a real Strömgren sphere?
It depends steeply on the star and the gas. A single O6 star (Q ≈ 10⁴⁹ photons/s) in gas of density n = 100 cm⁻³ gives R_S ≈ 3 parsecs. In diffuse gas (n = 1 cm⁻³) the same star ionizes a bubble tens of parsecs across. A hot B0 star (Q ≈ 10⁴⁶) in dense n = 1000 cm⁻³ gas produces a compact sphere only ~0.1 pc across — the regime of ultracompact H II regions like those seen with the VLA and JWST.
Is a real H II region actually a perfect sphere?
No. The classic Strömgren sphere assumes a single star at the center of uniform, static gas — a deliberate idealization. Real regions are lumpy, and the ionized bubble is overpressured relative to the neutral surroundings, so it expands supersonically. The ionization front drives a shock, and after ~a million years the region grows well beyond the initial static Strömgren radius toward the Spitzer expansion radius. Champagne flows, dust, and multiple stars distort the shape further.
What is the difference between a Strömgren sphere and an H II region?
The Strömgren sphere is the theoretical construct: the equilibrium ionized volume around one star in idealized gas. An H II region is the real observed object — the glowing ionized nebula, like the Orion Nebula or the Rosette. The Strömgren radius is the zeroth-order prediction for how large that H II region should be. Observers use the theory in reverse: measuring the size and emission of an H II region constrains the ionizing luminosity of the embedded star(s).
Why is it called Case B recombination?
Case B counts recombinations to all levels except the ground state. Recombinations straight to the ground state (n=1) emit a fresh Lyman-continuum photon that is immediately reabsorbed nearby, so it does not remove a photon from the global budget — it just gets recycled. This 'on-the-spot' approximation lets you use the Case B coefficient α_B ≈ 2.6×10⁻¹³ cm³/s (at 10⁴ K) instead of the larger Case A value, which would double-count those recycled photons.