Hypersonic Reentry Capsule (3D)

Educational model. This page explains how an unpowered, non-lifting capsule returning from orbit or from the Moon is slowed by the atmosphere, how hot its nose gets and why radio contact is briefly lost. It is a point-mass textbook model, not a mission-design or flight-safety tool.

Ballistic trajectory

The capsule is a point mass with ballistic coefficient β = m/(CdA) in an exponential atmosphere ρ = ρ0 e−h/H (ρ₀ = 1.225 kg/m³, H = 7.2 km). With speed v and path angle γ below the local horizontal: dv/dt = −ρv²/(2β) − g sinγ, dγ/dt = (g/v − v/(R+h)) cosγ, dh/dt = −v sinγ, integrated with RK4 (g falls with height, the Earth is a sphere). There is no lift, so a too-shallow or too-fast entry simply skips back out; the page says so.

Allen–Eggers closed form

For a straight path without gravity the equation integrates exactly: v = v0 exp(−ρH / (2β sinγ)). The deceleration peaks where ρ = β sinγ/H, with amax = v0² sinγ / (2eH) and speed v0e−1/2, independent of the ballistic coefficient (which only sets the altitude of the peak). The page runs a second, gravity-free integration and compares it with these formulas; the full run differs mostly for fast, shallow entries where the curved Earth flattens the path.

Stagnation-point heating

Convective heating of the nose follows the Sutton–Graves correlation q = K √(ρ/Rn) v³ with K = 1.74×10⁻⁴ kg^1/2/m (Earth air, SI). It grows with √ρ and v³, so the maximum occurs earlier than the deceleration peak, at ρ = ρs/3, speed v0e−1/6, with qmax = K v0³ e−1/2 √(ρs/(3Rn)). A hot wall re-radiates, giving the radiative-equilibrium temperature Tw = (q/εσ)1/4. The ablative shield is modelled only by an effective heat of ablation (25 MJ/kg, density 270 kg/m³, ablation starting once the flux exceeds what radiation at 1800 K can carry away): the recession depth is a rough guide, not a thermal-protection design. Radiative heating, which dominates for lunar-return speeds, is not included.

Plasma and radio blackout

Behind the bow shock air dissociates and ionises. A radio wave of frequency f cannot pass through plasma whose electron density exceeds the cutoff nc = ε0 me (2πf)² / e² = 1.24×1016 fGHz² m−3 (about 6×10¹⁶ m⁻³ for S-band at 2.2 GHz). The electron density here is a deliberately crude estimate: a Saha equilibrium for the ionisation of nitric oxide (9.26 eV) at an assumed shock-layer temperature 0.85 v (K, v in m/s), pressure ρv², 3% ionisable species, multiplied by a (ρ/(ρ+3×10⁻⁶))² factor that stands in for the collisional lag at high altitude. It captures the order of magnitude and the trends (more speed and more density mean more electrons; higher frequency means a shorter blackout) but not real non-equilibrium chemistry, so do not use the blackout altitudes quantitatively. Real vehicles also see the plasma density vary strongly around the body and in the wake.

Shock stand-off

The glowing shock layer in the pictures sits at the Billig stand-off distance of a sphere in a hypersonic stream, Δ ≈ 0.143 Rn; its colour and brightness are drawn from the heat flux and are not a radiative-transfer calculation.

In-page checks

(1) The gravity-free numerical run reproduces the Allen–Eggers peak deceleration, the speed at the peak and the whole v(h) curve. (2) The same run reproduces the analytic peak heat flux and its altitude, and the wall temperature closes the radiative balance. (3) The plasma cutoff coefficient 1.24×10¹⁶ matches ε₀m_e(2πf)²/e² computed from the physical constants, the Saha quadratic is satisfied and electron density rises monotonically with speed.

Independent educational model of reentry physics; it is not affiliated with any space agency or manufacturer and contains no design data.

What to try

• Pick Orbital return, press Play and watch the radio link drop out: blackout starts when the plasma frequency passes the carrier.
• Raise the link frequency to 32 GHz: the blackout window shrinks but does not vanish.
• Change the ballistic coefficient: the peak deceleration stays the same but happens lower down.
• Make the entry shallower than 1.5 degrees at 11 km/s: with no lift the capsule skips back out.