This is a sedimentation-column model of the same iron-rain core-formation process as the 3D sim — the classic way settling under Stokes drag is studied in the lab: a vertical slice from the planet's surface (top) down through the growing metallic core (bottom), rather than a full 3D sphere. Where the 3D sim plugs droplets straight into the Stokes terminal-velocity formula, this model integrates each droplet's actual equation of motion (Newton's 2nd law with gravity, buoyancy and linear Stokes drag) and only reaches that same terminal velocity as a limit:
m dv/dt = Δρ·V·g − 6πη·r·v (F = ma, real force balance)
⇒ dv/dt = −(v − v∞)/τ (exponential relaxation)
τ = (2/9)·ρ_iron·r²/η (relaxation time constant)
v∞ = (2/9)·Δρ·g·r²/η (Stokes terminal velocity — identical
formula to the 3D companion sim)
Because this relaxation ODE is numerically stiff (τ is typically ~1–4 ms, far shorter than one animation frame), the sim advances velocity with the exact exponential solution v(t+dt) = v∞ + (v(t) − v∞)·e−dt/τ each frame — a naive forward-Euler step would diverge for a ratio this large. The near-instant convergence to v∞ is itself the numerical proof of why the 3D sim's "always at terminal velocity" shortcut is a safe approximation, rather than an assumption taken on faith.
The core layer grows by real volume conservation, but a uniform-cross-section column converts volume fraction into height fraction linearly (height fraction = volume fraction), unlike the 3D sim's sphere, where the same volume fraction produces a much thinner-looking radius fraction (∝ cube root of volume) — a genuine geometric difference between the two representations, not an arbitrary rescaling.
The sim also reports a live Reynolds number Re = ρsilicate·v∞·2r/η, which Stokes' law requires to stay ≪ 1. Across most of the slider range it does — but at large grain size combined with low viscosity (e.g. r ≈ 15 mm, η ≈ 1 Pa·s) Re climbs into the hundreds, genuinely outside where linear Stokes drag is valid. Neither this sim nor the 3D one corrects the formula there; the Reynolds readout exists so that departure from validity is visible rather than hidden.
- Grain radius — droplet radius r in both the relaxation-time and terminal-velocity formulas; velocity grows with r², so doubling grain size quadruples settling speed (and moves Re further from the Stokes-valid regime).
- Viscosity η — a fully molten magma ocean is ~1–100 Pa·s; higher viscosity slows settling and shrinks Re, keeping the model safely in the Stokes regime.
- Surface gravity g — from Mercury-like (3.7 m/s²) to a super-Earth (24 m/s²); stronger gravity speeds up both settling and core growth.
- Playback speed — rescales only the animation clock (via the same model-time acceleration as the 3D sim), never the underlying physics.
Real-world relevance: this iron-rain mechanism is the leading model for how Earth, Mercury and Mars formed their metallic cores within roughly the first 10–100 million years after accretion, while their mantles were still fully molten.