Chapter 7 — Key Takeaways (Atmospheric Re-Entry)

A one-page reference. Reread this before an exam, or before you design any entry vehicle.

The one idea

Re-entry disposes of orbital energy as heat, on purpose. You refuse to pay the delta-v to brake with rockets (~$7.8\ \text{km/s}$, unaffordable) and instead spend a ~$100\ \text{m/s}$ deorbit burn to let the atmosphere brake you for free — converting ~$30\ \text{MJ/kg}$ of kinetic energy into a shock layer at ~$10{,}000\ \text{K}$, and surviving it with a shield.

Key equations (with symbols and units)

Equation Meaning Symbols
$\dfrac{E_k}{m} = \tfrac12 v^2$ kinetic energy to dissipate $v$ = entry speed (m/s); result J/kg
$\Delta T \approx \dfrac{v^2}{2c_p}$ ideal stagnation temperature rise $c_p \approx 1{,}005\ \text{J/(kg·K)}$ (air)
$\dot q \propto \sqrt{\dfrac{\rho}{R_n}}\,v^3$ stagnation heat flux (Sutton–Graves scaling) $\rho$ density, $R_n$ nose radius, $v$ speed
$\dfrac{D}{m} = \dfrac{\rho v^2}{2\beta}$ drag deceleration $\beta = m/(C_dA)$ ballistic coefficient (Ch. 5)
$a_{\max} = \dfrac{v_E^2 \sin\gamma_E}{2eH}$ peak deceleration (Allen–Eggers, ballistic) $\gamma_E$ entry angle, $H$ scale height, $e=2.718$
$f_p \approx 8.98\sqrt{n_e}$ Hz plasma frequency (blackout below it) $n_e$ electron density (m$^{-3}$)

Numbers worth memorizing

  • Orbital kinetic energy: ~30 MJ/kg from LEO (~7 kg TNT per kg); ~60 MJ/kg from the Moon.
  • Deorbit burn from LEO: ~100 m/s (vs ~7.8 km/s to stop propulsively — mass ratio ~13.5).
  • Shock-layer gas: ~6,000–11,000 K (ideal calc gives an impossible ~30,000 K → dissociation caps it).
  • Heating grows as $v^3$; energy as $v^2$ — so faster entries are disproportionately worse.
  • Peak g $\propto v_E^2 \sin\gamma_E$, independent of the vehicle's mass/$\beta$.
  • Apollo corridor: ~1–2° wide, centered ~$-6.5°$; blackout ~3 minutes; blocked below ~3–9 GHz.
  • Typical peaks: shallow LEO capsule ~4 g; ballistic lunar return ~36 g (→ why Apollo flew lifting).
  • TPS mass: ~10–20% of entry mass (mass is the enemy).

Compression, not friction — the blunt body

  • Heat comes from compressing air in the shock layer, not rubbing. Hottest point = stagnation point (the nose), where air is most fully brought to rest.
  • A blunt body wins twice: (1) a detached bow shock dumps energy into gas that blows away in the wake; (2) large nose radius $R_n$ lowers $\dot q$ (which $\propto 1/\sqrt{R_n}$). Drag, the villain of ascent, is the hero of re-entry.

Thermal protection — three strategies

Strategy How it works Reusable? Real vehicle
Ablative shield erodes/chars/vaporizes; departing mass carries heat away + blocks convection mostly single-use Apollo (AVCOAT), Dragon (PICA-X)
Thermal tile low-density insulator; hot surface re-radiates, conducts almost nothing inward many flights Space Shuttle (silica tiles, RCC)
Transpiration coolant sweated through porous skin forms an insulating film reusable in principle rare; studied for Starship

Rule of thumb: fastest/hottest entry → ablative; reuse required → tiles.

Ballistic vs. lifting — decision aid

You want… Choose Because
simplicity, robustness, no active control ballistic but high peak g and concentrated heating
low peak g, cross-range, precision, wide corridor lifting ($L/D\approx0.3$–$0.4$ capsule, $\approx1$ Shuttle) spreads deceleration over time; steers within corridor
gentle deceleration, peak heating in thin air low $\beta$ (broad, light) decelerates high in the atmosphere
deep, fast penetration (warheads) high $\beta$ (dense, compact) knifes to low altitude; endures the heat

The corridor — two walls

  • Too steep → excessive g and heating → burn up / crush.
  • Too shallow → skip back out of the atmosphere (or overshoot + long heat soak).
  • Lift widens and steers within the corridor (roll lift up = don't dive; down = don't skip). A controlled skip (Orion/Artemis) turns the failure mode into a range-extending technique.

Common pitfalls

Pitfall Reality
"Re-entry heating is friction." It is mainly compression of shock-layer gas; the nose (stagnation point) is hottest.
"Just use retro-rockets to land." Costs ~7.8 km/s of delta-v — a second launch vehicle. The atmosphere is the only affordable brake.
"A sharp, streamlined shape cuts heating." Backwards: a blunt body sheds heat into the shock layer. Sharp bodies burn up.
"Enter shallow to minimize g." Too shallow → skip out (fail to land) and a longer heat soak (heavier shield). Stay in the corridor.
"A heavier vehicle pulls more g." Peak g is independent of mass/$\beta$; the heavy one just peaks lower in the atmosphere.
"Blackout means something broke." It is the plasma sheath reflecting radio below $f_p$; it ends as the vehicle slows and the sheath recombines.

Mission / project additions this chapter

  • MDR: added an Entry / Disposal note — does any part of your mission enter an atmosphere, at what speed, and with what TPS and ballistic-coefficient class? (Track B: no atmosphere at the Moon → fully propulsive descent; Track D sample return: fastest, hottest Earth entry.)
  • Helper code (standalone, not a core astrotools module): peak_g(v_entry, gamma_deg, H), entry_angle_for_g(g_limit, v_entry, H), plasma_frequency(n_e), specific_ke(v).