Strömgren Sphere

A Strömgren sphere is a fundamental theoretical model in astrophysics—formulated by Danish astronomer Bengt Strömgren in 1937 and 1939—describing the spherical volume of fully ionized hydrogen (H II region, with temperature T ≈ 8,000–10,000 K and ionization fraction x = n_p / n_H ≈ 1) created when a hot, luminous young star (spectral type O or early B, with effective surface temperature T_eff ≈ 25,000–50,000 K) is embedded within a homogeneous, quiescent cloud of neutral atomic hydrogen (H I).

Within the Strömgren sphere, the central star's total output rate of hydrogen-ionizing Lyman-continuum (LyC) ultraviolet photons (Q(H⁰) or N_LyC, with photon energies hν ≥ I_H = 13.6 eV or wavelengths λ ≤ 91.2 nm) is exactly balanced by the integrated volumetric rate of radiative proton–electron recombinations into excited states of neutral hydrogen across the interior volume:

R_S = [ 3 Q(H⁰) / (4π α_B(T) n_H²) ]^1/3

where R_S is the Strömgren radius, n_H ≈ n_p ≈ n_e is the number density of hydrogen nuclei (protons) and free electrons inside the ionized region, and α_B(T) ≈ 2.59 × 10⁻¹³ (T / 10⁴ K)^−0.7 cm³ s⁻¹ is the Case B radiative recombination coefficient (Osterbrock & Ferland, 2006). Beyond observational H II nebulae such as the Orion Nebula (M42), the Rosette Nebula, and giant extragalactic H II regions in the Pinwheel Galaxy, overlapping cosmological Strömgren spheres around the first Population III stars, starburst galaxies, and quasars govern the Epoch of Reionization (z ≈ 15 → 6) traced by the Lyman-alpha forest.

1. Photoionization vs. Radiative Recombination Balance

When an extreme-ultraviolet photon with energy hν ≥ 13.6 eV strikes a ground-state neutral hydrogen atom H(1s), it photoionizes the atom (H⁰ + γ → p⁺ + e⁻) with a threshold cross-section at λ = 91.2 nm of:

σ₀ = 6.30 × 10⁻¹⁸ cm² (scaling above threshold as σ(ν) ≈ σ₀ (ν / ν₀)⁻³)

The ejected photoelectron carries away excess kinetic energy E_k = hν − 13.6 eV, which is rapidly shared via elastic Coulomb collisions with ambient electrons and protons, heating the H II plasma to T ≈ 8,000–10,000 K (thermostatted by collisionally excited forbidden-line cooling of trace O⁺⁺ and N⁺ ions, such as [O III] 495.9, 500.7 nm).

Simultaneously, free electrons are recaptured by protons into quantum level n of neutral hydrogen via radiative recombination (p⁺ + e⁻ → H(n) + γ) at a volumetric rate per unit volume of:

r_rec = n_e n_p α(T) ≈ x² n_H² α(T) (cm⁻³ s⁻¹)

where α(T) = ∑ α_n(T) is the total recombination coefficient summed over principal quantum numbers n.

2. The "On-the-Spot" Approximation (Case B Recombination)

In 1938, James G. Baker and Donald H. Menzel distinguished two optical-depth regimes for radiative recombination:

• Case A Recombination (Optically Thin to Lyman Continuum, α_A = ∑_n=1^∞ α_n ≈ 4.18 × 10⁻¹³ cm³ s⁻¹ at 10⁴ K): Includes direct recombinations into the n = 1 ground state. However, whenever a free electron recombines directly into the n = 1 ground state (accounting for ~38% of all recombinations), it immediately emits a new diffuse Lyman-continuum photon (hν ≥ 13.6 eV).
• Case B Recombination (Optically Thick "On-the-Spot" Approximation, α_B = ∑_n=2^∞ α_n ≈ 2.59 × 10⁻¹³ cm³ s⁻¹ at 10⁴ K): In any realistic interstellar H II region, the optical depth to ground-state ionizing photons is large, so a diffuse n = 1 recombination photon travels only a short distance before ionizing another nearby H⁰ atom—resulting in zero net loss of ionizing photons. Only recombinations to excited states (n ≥ 2)—which cascade downward by emitting Balmer (Hα, Hβ), Paschen, and Lyman-alpha (hν = 10.2 eV < 13.6 eV) photons that cannot re-ionize ground-state hydrogen—permanently remove an ionizing photon from the budget.

Equating the star's total ionizing photon emission rate Q(H⁰) (photons s⁻¹) to the total Case B recombination rate inside a sphere of radius R_S with x ≈ 1 (n_e = n_p = n_H):

Q(H⁰) = (4/3) π R_S³ n_e n_p α_B(T) = (4/3) π R_S³ n_H² α_B(T)

yields the classic Strömgren radius:

R_S = [ 3 Q(H⁰) / (4π α_B(T) n_H²) ]^1/3 ≈ 3.15 pc × (Q(H⁰) / 10⁴⁹ s⁻¹)^1/3 × (n_H / 100 cm⁻³)^−2/3

A striking prediction of Strömgren's 1939 paper is that the ionization fraction x(r) does not decline gradually with radius as 1/r²; instead, the gas remains >99.9% ionized throughout almost the entire interior of the sphere (r < 0.95 R_S) and then drops abruptly from x ≈ 1 to x ≈ 0 across a thin spherical shell (the ionization front) whose thickness ΔR equals roughly one to two photon mean free paths in neutral hydrogen:

ΔR ≈ λ_mfp = 1 / (n_H σ₀) ≈ 1 / [ n_H (6.3 × 10⁻¹⁸ cm²) ] ≈ 0.05 parsecs / (n_H / 1 cm⁻³)

In a typical giant molecular cloud core with n_H = 100 cm⁻³ surrounding an O5 V star (R_S ≈ 4.3 pc), the transition thickness is only ΔR ≈ 0.0005 pc (≈ 100 AU)—meaning ΔR / R_S ≈ 10⁻⁴! Physically, as soon as the neutral fraction (1 − x) rises above a few tenths of a percent near r ≈ R_S, the huge photoionization cross-section σ₀ causes optical depth τ = ∫ (1 − x) n_H σ₀ dr to climb above unity over just 100 AU, extinguishing all remaining UV photons and producing the crisp outer rims observed in Hubble and James Webb Space Telescope images of H II regions.

Because only the Wien tail of a stellar blackbody spectrum lies shortward of the 91.2 nm Lyman edge (hν / k_BT_eff = 157,800 K / T_eff), the ionizing photon luminosity Q(H⁰) and Strömgren radius R_S drop precipitously from early O stars to B and A stars (Martins et al., 2005; Osterbrock & Ferland, 2006):

From this relation also follows the Kennicutt (1998) Hα Star Formation Rate Calibration: because every Case B recombination produces an average of 0.45 Hα photons (λ = 656.3 nm), the total Hα luminosity L(Hα) of an unreddened Strömgren sphere is directly proportional to Q(H⁰) (L(Hα) [erg s⁻¹] ≈ 1.37 × 10⁻¹² Q(H⁰) [s⁻¹]), allowing astronomers to measure the birth rate of massive O/B stars in distant galaxies directly from their integrated Hα emission.

When a massive O star ignites inside a cold molecular cloud (T₀ ≈ 10–100 K), the Strömgren sphere evolves through two distinct hydrodynamic phases (Kahn, 1954; Spitzer, 1978):

1. Phase 1 — Supersonic R-Type ("Rarefied") Ionization Front (t < t_rec ≈ 1 / [n_H α_B] ≈ 1,200 yr at n_H = 100 cm⁻³): Initially, stellar UV photons stream out into the neutral gas far faster than the speed of sound (v_IF = Q(H⁰) / (4πr²n_H) ≫ c_s ≈ 10 km s⁻¹). The gas has no time to move hydrodynamically; it is simply flash-ionized and heated in place from 100 K to 10,000 K at constant density n₀ until the front decelerates to ~2c_s at the initial Strömgren radius R_S,0.
2. Phase 2 — Subsonic D-Type ("Dense") Pressure-Driven Spitzer Expansion: Once the ionization front reaches R_S,0, the newly ionized interior has the same number density as the surrounding cold cloud (n_i = n₀), possesses twice as many particles per unit volume (protons + electrons vs. neutral H atoms), and is 100 times hotter (10,000 K vs. 100 K). By the ideal gas law, the thermal pressure inside the Strömgren sphere (P_i = 2n₀k_BT_i) is ~200 times higher than the ambient cold gas pressure! This overpressure drives a supersonic shock wave into the surrounding neutral cloud ahead of a subsonic D-type ionization front, sweeping up a dense neutral shell (often triggering second-generation star formation via the "collect and collapse" mechanism) while the H II region radius R_i(t) expands according to Lyman Spitzer's (1978) 4/7-power law:

R_i(t) = R_S,0 [ 1 + (7/4) (c_i t / R_S,0) ]^4/7

where c_i ≈ 10 km s⁻¹ is the isothermal sound speed in the ionized gas.

On cosmological scales (1 to 50 comoving megaparsecs), generalized expanding Strömgren spheres (Shapiro & Giroux, 1987) describe the Epoch of Reionization (z ≈ 15 → 6), when the first stars and active galactic nuclei ionized the neutral intergalactic medium (IGM) formed after Cosmic Microwave Background recombination (z ≈ 1100):

• Around z ≈ 12–10, isolated early starburst galaxies and quasars carved out individual ionized H II "bubbles" (cosmological Strömgren spheres) inside a predominantly neutral intergalactic medium—directly observed via damped Lyman-alpha damping wings in high-redshift JWST galaxy spectra and via the quasar near-zone effect at z > 6 (Fan et al., 2006).
• Around z ≈ 7–5.5 (the Overlap Phase), neighboring Strömgren bubbles percolated and merged throughout the universe, raising the photon mean free path by an order of magnitude and leaving the intergalactic medium >99.99% ionized to the present day.

1. Strömgren, B. (1939). "The Physical State of Interstellar Hydrogen." The Astrophysical Journal, 89, 526–547. https://doi.org/10.1086/144074
2. Kahn, F. D. (1954). "The acceleration of interstellar clouds." Bulletin of the Astronomical Institutes of the Netherlands, 12(456), 187–202. https://ui.adsabs.harvard.edu/abs/1954BAN....12..187K
3. Spitzer, L. (1978). Physical Processes in the Interstellar Medium. John Wiley & Sons. https://doi.org/10.1002/9783527617722
4. Shapiro, P. R., & Giroux, M. L. (1987). "Cosmological H II regions and the photoionization of the intergalactic medium." The Astrophysical Journal Letters, 321, L107–L112. https://doi.org/10.1086/185015
5. Osterbrock, D. E., & Ferland, G. J. (2006). Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (2nd ed.). University Science Books. https://uscibooks.aip.org/books/astrophysics-of-gaseous-nebulae-and-active-galactic-nuclei/