Strömgren Disk: 2D Ionization Front Growth

Ionizing Star

UV photon rate Q 1.0×10⁴⁹ s⁻¹

Gas density n 100 cm⁻³

Slab thickness L 0.30 pc

Simulation speed 1.0×

Ionization Front

Equilibrium radius R_s

— pc

Current front R(t)

— pc

Recombination time

— yr

Elapsed time

0 yr

Ionized area fraction

0.0%

ODE residual at front |dR²/dt|

—

How it works (2D areal balance)

This is a genuine 2D reduction of the ionization-balance problem — a "Strömgren disk" — not a flattened picture of the 3D sphere. The star sits inside a thin slab of hydrogen gas of fixed thickness L (like a galactic disk or a compact molecular clump). Because the gas can only recombine where it physically exists, the ionized region is a cylinder of radius R and height L, not a sphere: recombinations scale with cylinder volume π R² L, not spherical volume (4/3)π R³. Photon conservation for that geometry gives:

d/dt[ π R² L n ] = Q − π R² L n² α_B
⇒ 2π L n R·dR/dt = Q − π L n² α_B R²

Substituting x = R² turns this into a linear ODE that this simulation integrates numerically every frame (RK4, not a pre-baked formula):

dx/dt = Q/(π n L) − n α_B x        (x = R²)

equilibrium:  R_s = √( Q / (π n² α_B L) )
time-dependent:  R(t) = R_s · √( 1 − e^(−t/t_rec) )
t_rec = 1 / (n α_B)

Notice the exponent: the disk's front grows as t^1/2 near ignition, while the real 3D Strömgren sphere grows as t^1/3 — a direct, testable signature of confining the same photon-balance physics to a fixed-height slab instead of letting it expand freely in 3D. The slab thickness L also enters the equilibrium radius directly: a thinner slab concentrates less gas per unit area, so the front must reach farther in R before recombination catches up with Q.

UV photon rate Q — the star's ionizing photon output; higher Q pushes the equilibrium radius outward.
Gas density n — denser gas recombines faster: it shrinks R_s and shortens t_rec, since more available electrons/protons collide sooner.
Slab thickness L — how tall the confining gas layer is; a thinner slab forces the front to spread wider in radius to soak up the same photon budget.
Simulation speed — plays the RK4 integration through many recombination times; the "ODE residual" readout confirms the numerical front tracks the analytic equilibrium (residual → 0 as t → ∞).
Pink dots are ionized (H⁺); brown dots are neutral hydrogen (H I) still waiting for the front to reach them.

Ionized (H⁺) Neutral (H I) Ionizing star