A returning booster falls tail-first through thickening air. Four lattice ("grid") fins near the top bite into the airstream and rotate to steer — no propellant used until the final landing burn. The air gets denser as altitude drops:
ρ(h) = ρ₀ · e^(−h/H) ρ₀ = 1.225 kg/m³, H = 8500 m
q = ½ ρ(h) v² dynamic pressure ("max-Q" region is where q peaks)
F_fin = q · A_fin · Clα · δ side force from fin deflection δ
a_x = F_fin / m lateral acceleration steering cross-range
Because q depends on both altitude and speed squared, fin authority is not constant: high in the thick lower atmosphere while still moving fast, but it collapses toward zero as the vehicle slows near touchdown — real guidance software front-loads its steering commands for exactly this reason. Crosswind adds its own q-scaled lateral push that the fins must cancel out.
- Fin deflection slider — manual steering command δ, in degrees, sent to all four fins together (differential yaw is simplified to a single 2D lateral channel here).
- Crosswind — a steady lateral gust the fins have to fight; its force also scales with q, so it bites hardest exactly when the booster is deepest in the atmosphere.
- Autopilot — a proportional-derivative guidance law, δ = −(K_p·x + K_d·v_x) clamped to ±20°, that nulls cross-range error the way real flight software does.
- q gauge — normalized dynamic pressure; watch fin authority (how fast cross-range error responds to deflection) shrink as the bar empties near the ground.
Real-world relevance: this is the aerodynamic entry-and-descent phase SpaceX's Falcon 9 and Falcon Heavy boosters fly between the entry burn (~70 km) and the landing burn (~1 km) — grid fins were the enabling innovation that made routine booster reuse geometrically precise enough to land on a droneship or launch pad.