Engineering & Science
Strömgren Sphere Radius (RS) & H II Nebula Recombination Timescale
Calculate the ionization-bounded H II region Strömgren radius RS = ((3 * Q_H) / (4 * PI() * n_H^2 * alpha_B))^(1/3) in parsecs and hydrogen recombination time t_rec = 1 / (n_H * alpha_B) in years.
Balance O-star Lyman-continuum UV photons against electron-proton recombination!
Hot young O- and B-type stars (like Theta-1 Orionis C in the Orion Nebula) flood surrounding molecular gas with ionizing Lyman-continuum photons (E >= 13.6 eV) at rate Q_H (photons/s)! Bengt Strömgren (1939) showed that ionization carves out a sharply bounded spherical H II region of radius R_S where total ionizing photon output Q_H exactly balances Case B radiative recombinations throughout the sphere's volume: (4/3) * PI() * (R_S^3) * (n_H^2) * alpha_B = Q_H!
Solving for R_S in CGS units and dividing by 1 parsec = 3.0856776e18 cm gives R_S_pc = (((3 * Q_H) / (4 * PI() * (n_H^2) * alpha_B)) ^ (1/3)) / 3.0856776e18! For an O6V star (Q_H = 1.0e49 photons/s) in interstellar gas of density n_H = 100 cm^-3 at 10,000 K (Case B recombination coefficient alpha_B = 2.6e-13 cm^3/s), the Strömgren sphere spans R_S = 3.150 parsecs (~10.3 light-years) with recombination time t_rec = 1 / (n_H * alpha_B) = 1,218.8 years; in a 10x denser core (n_H = 1,000 cm^-3), R_S shrinks by 10^(2/3) = 4.64x to 0.679 pc!
| Nebular Gas Density n_H (Q_H = 10^49 s^-1) | Strömgren Radius R_S (pc) | Recombination Time t_rec (yr) | Density Scaling |
|---|---|---|---|
| Classic H II Region (n_H = 100 cm^-3) | 3.150 parsecs | 1,218.8 years | Baseline Orion-like nebula |
| Compact H II Core (n_H = 1,000 cm^-3) | 0.679 parsecs | 121.9 years | R_S scales as n_H^(-2/3) |
| Invalid Density n_H <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid Nebula Inputs" |
Strömgren Sphere Radius R_S in Parsecs (pc)
Evaluates CGS cube root (((3 * Q_H) / (4 * PI() * n_H^2 * alpha_B))^(1/3)) and converts centimeters to parsecs (/ 3.0856776e18).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Nebula Inputs", ROUND((((3 * B2) / (4 * PI() * (B3 ^ 2) * B4)) ^ (1 / 3)) / 3085677600000000000, 3))Hydrogen Recombination Timescale t_rec = 1 / (n_H * alpha_B) in Julian Years
Divides seconds 1 / (n_H * alpha_B) by 31,557,600 seconds per Julian year.
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Nebula Inputs", ROUND((1 / (B3 * B4)) / 31557600, 1))Try the example
Use the original values or change them to test your understanding.