Reentry Velocity⚠ unverified

Understanding the Parameters

• Entry velocity $V_0$ — the speed at the atmospheric "interface" (conventionally $\sim 122\,\text{km}$ at Earth). Low-Earth-orbit return is $\sim 7.8\,\text{km/s}$; lunar return $\sim 11\,\text{km/s}$; interplanetary return higher still. The whole profile scales with $V_0$.

• Ballistic coefficient $\beta$ — the vehicle's resistance to deceleration (Ballistic Coefficient). It sits in the denominator of the exponent, so a high $\beta$ makes the argument small — the vehicle barely slows until deep in the atmosphere, then decelerates violently at low altitude. A low $\beta$ slows the vehicle gently and high up.

• Flight-path angle $\gamma$ — the steepness of the descent below horizontal. A steep entry (large $\sin\gamma$) shrinks the exponent: the vehicle traverses the atmosphere quickly, decelerating hard and low (high $g$, high peak heat flux, short duration). A shallow entry spreads deceleration over a long, gentle, hot-for-longer path — and too shallow, the vehicle skips back out (Skip Reentry Delta V). The registry labels $\gamma$ dimensionless, but it is degrees (the code converts internally).

• Scale height $H$ and $\rho_0$ — describe the exponential atmosphere $\rho = \rho_0 e^{-h/H}$ ($H \approx 7$–$8\,\text{km}$ for Earth). The grouping $\rho_0 H/(2\beta\sin\gamma)$ sets the total deceleration.

• The double-exponential shape — as $h$ falls, $e^{-h/H}$ grows, and $V$ drops as the exponential of that. The result is a sharp "knee": most of the speed is lost within roughly one scale height, which is why entry heating and $g$-loads are concentrated in a brief, intense pulse.

Derivation (Approaching a Proof)

Allen and Eggers made three assumptions: (1) a purely ballistic (non-lifting) entry, (2) an exponential atmosphere $\rho = \rho_0 e^{-h/H}$, and (3) that during the intense deceleration phase, drag dominates gravity and the flight-path angle $\gamma$ stays roughly constant. The equation of motion along the trajectory is then just drag decelerating the vehicle:

$$m\frac{dV}{dt} = -D = -\tfrac12\rho V^2 C_d A = -\frac{\rho V^2}{2}\cdot\frac{m}{\beta} \;\Longrightarrow\; \frac{dV}{dt} = -\frac{\rho V^2}{2\beta}.$$

Convert time to altitude using the geometry $dh = -V\sin\gamma\,dt$ (the vehicle loses altitude at rate $V\sin\gamma$):

$$\frac{dV}{dh} = \frac{dV}{dt}\cdot\frac{dt}{dh} = \frac{-\rho V^2/(2\beta)}{-V\sin\gamma} = \frac{\rho V}{2\beta\sin\gamma}.$$

Separate variables and substitute $\rho = \rho_0 e^{-h/H}$:

$$\frac{dV}{V} = \frac{\rho_0 e^{-h/H}}{2\beta\sin\gamma}\,dh.$$

Integrate from the entry state ($h \to \infty$, $V = V_0$) down to altitude $h$. The right side integrates to $-\dfrac{\rho_0 H}{2\beta\sin\gamma}e^{-h/H}$, giving

$$\ln\frac{V}{V_0} = -\frac{\rho_0 H}{2\beta\sin\gamma}\,e^{-h/H} \;\Longrightarrow\; V(h) = V_0\,\exp\!\left[-\frac{\rho_0 H}{2\beta\sin\gamma}\,e^{-h/H}\right]. \qquad\blacksquare$$

The nested exponential is exact within the model. Differentiating it yields the deceleration (Reentry Deceleration) and, combined with the heating law, the peak-heating altitude (Peak Heating Altitude) — all in closed form, which is why the Allen–Eggers solution remains the standard first-cut entry analysis.

Dimensional check. The exponent's group $\dfrac{\rho_0 H}{2\beta\sin\gamma} = \dfrac{(\text{kg}/\text{m}^3)(\text{m})}{(\text{kg}/\text{m}^2)} = \text{(–)}$ is dimensionless (as an argument of $\exp$ must be), so $V$ carries the units of $V_0$, m/s. $\checkmark$

History and Development

• Allen and Eggers, 1958. H. Julian Allen and Alfred J. Eggers of NACA derived this solution and, with it, the counter-intuitive blunt-body conclusion: a low-$\beta$, high-drag shape decelerates high in the thin upper atmosphere, dumping most of the entry energy into the air rather than the vehicle. It made survivable crewed re-entry possible and shaped every capsule from Mercury to Orion.

• A closed-form gem. That a nonlinear entry trajectory collapses to a clean analytic velocity profile is remarkable and pedagogically central; it gives the peak-deceleration result $a_{max} = V_0^2\sin\gamma/(2eH)$, famously independent of ballistic coefficient (Reentry Deceleration).

• Limits of the model. The assumptions break down for lifting entry (Shuttle, Apollo's offset CG), shallow skip trajectories, and where gravity and $\gamma$-variation matter. Modern entry design integrates the full equations numerically, but Allen–Eggers remains the sanity check and the source of physical intuition.

Related Concepts: Ballistic Coefficient, Reentry Deceleration, Peak Heating Altitude, Reentry Time, Stagnation Heat Flux, Reentry Range, Skip Reentry Delta V, Drag Force

Notes: Registry calculator reentry-velocity (unverified). Allen–Eggers ballistic-entry velocity profile — correct as shipped. Assumes non-lifting entry, exponential atmosphere, constant $\gamma$, drag ≫ gravity. $\gamma$ (gamma_deg) is degrees despite the dimensionless label. Peak deceleration $a_{max}=V_0^2\sin\gamma/(2eH)$ is independent of $\beta$.