Reentry Time⚠ unverified

Understanding the Parameters

• Scale height $H$ — the vertical distance over which atmospheric density changes by a factor $e$ ($\sim 7$–$8\,\text{km}$ for Earth). Because entry heating and deceleration are concentrated within about one scale height, $H$ sets the length of the intense phase, and the characteristic time is how long the vehicle spends crossing it.

• Entry velocity $V_0$ and angle $\gamma$ — the vehicle descends vertically at rate $V_0\sin\gamma$. A faster or steeper entry crosses the scale height sooner, so the intense phase is shorter (but more violent — higher peak $g$ and flux). A shallow entry is longer and gentler, spreading the heat load over more time. The registry omits $\sin\gamma$; the consistent form includes it.

• Ballistic coefficient $\beta$ — appears in the registry formula but not in the physical descent-time estimate; the characteristic time is a kinematic quantity (how fast you cross one scale height), independent of the vehicle's drag properties to first order.

• The output $t$ — a characteristic duration, not an exact entry time. Real entries: the fiery phase of a capsule lasts on the order of a minute; the Space Shuttle's much shallower, lifting entry stretched the thermal soak to tens of minutes, deliberately trading a longer, cooler exposure for lower peak flux.

Derivation (Approaching a Proof)

The intense deceleration and heating of entry are concentrated within roughly one atmospheric scale height $H$ around the peak (the density — and hence drag and heating — falls off by $e$ per scale height, so contributions from higher and lower altitudes fade quickly). A natural time scale is therefore how long the vehicle takes to descend one scale height.

The vehicle loses altitude at the vertical component of its velocity, $\dot h = V\sin\gamma$. Taking the entry speed $V_0$ and a roughly constant flight-path angle $\gamma$ (the Allen–Eggers assumption, Reentry Velocity), the time to traverse one scale height is

$$t \approx \frac{H}{\dot h} = \frac{H}{V_0\sin\gamma}. \qquad\blacksquare$$

For a steep entry ($\gamma = 90^\circ$, $\sin\gamma = 1$) at $V_0 = 7\,\text{km/s}$ with $H = 7\,\text{km}$, this gives $t \approx 1\,\text{s}$ per scale height — the deceleration pulse is only a few such times wide, hence the "tens of seconds" intensity of a steep ballistic entry. Shallow entries ($\sin\gamma \ll 1$) stretch this out by $1/\sin\gamma$, the basis for the long, gentle, lifting entries that keep peak heating manageable.

Dimensional check (consistent form). $$\left[\frac{H}{V_0\sin\gamma}\right] = \frac{\text{m}}{\text{m}/\text{s}} = \text{s}.\ \checkmark$$ The registry form $H/(\beta V_0) = \text{m}/[(\text{kg}/\text{m}^2)(\text{m}/\text{s})] = \text{m}^2\text{s}/\text{kg}$ — the dimensional flag above.

History and Development

• The brevity of entry. That atmospheric entry is a transient — a brief, intense pulse rather than steady heating — is one of its defining features, and it follows directly from the exponential atmosphere: everything happens within a scale height or two. Allen–Eggers theory (Reentry Velocity) makes this quantitative.

• Trading duration for intensity. Entry design chooses where to sit on the duration–intensity trade. Steep ballistic entries (warheads, sample-return capsules) are short and fierce; shallow lifting entries (Shuttle, crewed capsules) are long and mild. The characteristic time is the knob, set through the flight-path angle.

• Transient heat shields. Because the heat pulse is short, ablative and high-heat-capacity shields work by surviving a transient — absorbing and rejecting energy for a minute or two — rather than reaching steady state, which is why the total heat load (Reentry Heat Load) and duration matter as much as the peak flux.

Related Concepts: Reentry Velocity, Ballistic Coefficient, Reentry Deceleration, Peak Heating Altitude, Reentry Heat Load, Reentry Range

Notes: Registry calculator reentry-time (unverified). Registry form $H/(\beta V_0)$ is dimensionally inconsistent (yields m²·s/kg, not seconds) and injects $\beta$; the consistent estimate is $t\approx H/(V_0\sin\gamma)$ — the time to descend one scale height. Entry's intense phase is a short transient. Flagged in Known Issues.