Atmospheric entry - 2D trajectory and heating

Equations of motion

For a capsule of ballistic coefficient β = m/(C_dA) in an exponential atmosphere ρ = ρ₀e^-h/H (H = 7.2 km): dv/dt = -ρv²/(2β) + g sinγ and dγ/dt = (g/v - v/r) cosγ - (L/D) ρv/(2β), with γ measured below the horizon and r = R_E + h. The simulation integrates this with RK4.

Allen-Eggers solution

For a straight path and no gravity, v = v₀ exp(-ρH/(2β sinγ)). Peak deceleration is a_max = v₀² sinγ /(2eH) and does not depend on β at all; β only sets the altitude of the peak, h = H ln(ρ₀H/(β sinγ)). Peak stagnation heating (flux ∝ √ρ v³) occurs at v = v₀ e^-1/6 = 0.846 v₀.

Heating and plasma

Stagnation-point convective flux uses the Sutton-Graves correlation q = 1.7415×10^-4 √(ρ/R_n) v³ (SI units). A blunt nose (large R_n) lowers heating. The glow and wake drawn on the capsule scale with q: in reality the shock layer is ionised above roughly 3 km/s, which is the plasma sheath.

Honest simplifications: no radiative heating (important above about 10 km/s), no real-gas chemistry, no ablation, a fixed isothermal exponential atmosphere, a point-mass capsule with a constant lift-to-drag ratio, and no communication-blackout model. Real heat shields are designed with far more detailed codes.

• "LEO capsule return": watch the g-load peak, then compare the simulated curve with the dashed Allen-Eggers curve in the first graph.
• "Steep ballistic": much higher g-load and peak flux in a short time.
• "Lunar return": faster entry, larger heat load; raise the nose radius to cut flux.
• "Shallow skip attempt": with lift the capsule can bounce back out of the atmosphere.