Coming home: reentry

Every astronaut comes home the same way: by turning speed into heat. A capsule arriving from orbit carries enough energy per kilogram to melt its own weight in steel many times over. The whole art of reentry is getting rid of that energy slowly enough to survive, and quickly enough not to bounce back into space.

The engines start the fall. The air does the braking.

Look

1. The deorbit burn. Pointing backwards, the capsule fires its engine for a few minutes and slows down by about 100 m/s. That lowers the far side of its orbit into the atmosphere.
2. Entry interface, about 120 km up. The capsule meets the first thin traces of air still moving at about 7.8 km/s.
3. The fireball. Air ahead of the heat shield is squeezed so hard it becomes glowing plasma, thousands of degrees hot. Radio signals cannot get through for a few minutes. The heat shield slowly chars and flakes away, carrying the heat with it.
4. The corridor. Returning from the Moon at 11 km/s, the entry angle must be right to a fraction of a degree: too steep and the deceleration would crush the crew, too shallow and the capsule skims off the air like a stone off water.
5. Parachutes. Below about 10 km the capsule is falling at a few hundred km/h. Parachutes open, and it lands at a few metres per second.

Understand

From a 400 km circular orbit (r = 6,778 km) the speed is 7.67 km/s. An ellipse whose low point is 50 km up has a = 6,603 km, and at 400 km its speed must be 7.57 km/s. The deorbit burn is the difference: about 102 m/s, just 1.3% of the orbital speed.

Drag force is ½ ρ v² C_D A. Air density ρ roughly halves every 5 km of descent (scale height about 7.2 km), so the deceleration rises steeply, peaks, and falls as the capsule slows. In the model above:

Entry	Speed	Angle	Peak deceleration
From low orbit	7.85 km/s	1.5°	about 8 g
From the Moon, too steep	11 km/s	8°	about 31 g
From the Moon, in the corridor	11 km/s	5.6°	about 7 g
From the Moon, too shallow	11 km/s	4.8°	skips back out

Real capsules fly with a little lift (their centre of mass is off-centre, so they fly tilted) and roll to steer it, which lowers the peak to about 4 g for a Soyuz and widened Apollo’s corridor to about 2 degrees.

Master

The model integrates the planar equations of motion with gravity and drag only:

dv/dt = −μ r/|r|³ − (ρ(h) |v| / 2β) v,   ρ(h) = ρ₀ e^(−h/H)

with β = m/(C_D A) = 350 kg/m² (capsule-like) and H = 7.2 km. For a steep ballistic entry, Allen and Eggers (1958) showed the peak deceleration is nearly independent of β:

a_max ≈ v_E² sin γ / (2 e H)

For the steep lunar case (γ = 8°) it gives about 44 g; the full simulation gives 31 g, because gravity keeps bending the path shallower as the capsule descends. Either way, the entry angle, not the vehicle’s size, sets the g-load, which is why every returning spacecraft flies a precise corridor.