Peak Heating Altitude⚠ unverified

Understanding the Parameters

• Ballistic coefficient $\beta$ — a high-$\beta$ (dense, slender) vehicle carries its speed deeper before decelerating, so it peaks in hotter, denser air lower down; a low-$\beta$ (blunt) vehicle sheds speed high up and peaks higher, in thinner air. In the correct form $\beta$ is in the denominator of the log argument, so $h_q$ decreases with $\beta$ — the opposite of the registry's placement.

• Scale height $H$ — sets the vertical scale of the atmosphere. It multiplies the logarithm (so a thicker atmosphere raises the peak-heating altitude) and appears inside it.

• Flight-path angle $\gamma$ (correct form) — a steeper entry (larger $\sin\gamma$) drives the peak lower (the vehicle penetrates fast into dense air before slowing), while a shallow entry peaks higher. The registry form omits this dependence.

• The output $h_q$ — the altitude where the heat shield sees maximum flux. For Earth orbital entry it typically falls around $50$–$70\,\text{km}$. Peak heating altitude is higher than peak deceleration altitude (Reentry Deceleration) — heating peaks earlier because it scales with $\sqrt\rho\,V^3$ (more velocity-weighted) versus deceleration's $\rho V^2$.

Derivation (Approaching a Proof)

Stagnation-point convective heating follows $q \propto \sqrt{\rho}\,V^3$ (Stagnation Heat Flux). Insert the Allen–Eggers velocity profile $V = V_0\exp[-B\,e^{-h/H}]$ with $B = \rho_0 H/(2\beta\sin\gamma)$ (Reentry Velocity) and the exponential atmosphere $\rho = \rho_0 e^{-h/H}$:

$$q \propto \rho^{1/2}V^3 = \rho_0^{1/2}e^{-h/2H}\,V_0^3\exp\!\big[-3B\,e^{-h/H}\big].$$

Take the logarithm and differentiate with respect to $h$ to find the maximum:

$$\frac{d}{dh}\Big[-\frac{h}{2H} - 3B\,e^{-h/H}\Big] = -\frac{1}{2H} + \frac{3B}{H}e^{-h/H} = 0 \;\Longrightarrow\; e^{-h/H} = \frac{1}{6B}.$$

The peak-heating density is therefore $\rho^* = \rho_0 e^{-h/H} = \rho_0/(6B) = \dfrac{\beta\sin\gamma}{3H}$, and solving $e^{-h/H} = 1/(6B)$ for the altitude:

$$h_q = H\ln(6B) = H\ln\!\frac{6\rho_0 H}{2\beta\sin\gamma} = H\ln\!\frac{3\rho_0 H}{\beta\sin\gamma}. \qquad\blacksquare$$

Compare with peak deceleration, where $q \propto \rho V^2$ gives $e^{-h/H} = 1/(2B)$ and a lower altitude $h_a = H\ln(\rho_0 H/(\beta\sin\gamma))$. Because heating carries a higher power of velocity, its peak comes earlier (higher, faster) than the deceleration peak — a key fact in sequencing heat-shield and structural loads.

Dimensional check. The argument $3\rho_0 H/(\beta\sin\gamma) = (\text{kg}/\text{m}^3)(\text{m})/(\text{kg}/\text{m}^2) = \text{(–)}$ is dimensionless (as a log argument must be), and $H\ln(\cdots)$ has units of metres. $\checkmark$

History and Development

• Allen–Eggers heating analysis. The same 1958 ballistic-entry theory that gives the velocity profile yields closed-form peak-heating and peak-deceleration altitudes. The result that heating peaks higher than deceleration guided how heat shields and structures were sized against different points on the trajectory.

• Convective vs radiative. The $\sqrt\rho\,V^3$ law is for convective heating (Sutton–Graves, Stagnation Heat Flux); at the highest entry speeds a radiative term ($\propto \rho^{a}V^{b}$ with larger $b$) adds and can peak even higher and hotter, as for lunar and interplanetary returns.

• TPS sizing. The peak-heating point sets the maximum heat flux, while the integral over the trajectory sets the total heat load (Reentry Heat Load); together they size the thermal protection system (Thermal Protection Thickness).

Related Concepts: Reentry Velocity, Ballistic Coefficient, Reentry Deceleration, Stagnation Heat Flux, Reentry Heat Load, Thermal Protection Thickness, Reentry Heat Flux

Notes: Registry calculator peak-heating-altitude (unverified). Registry form $H\log(2\beta H)$ is non-standard — log argument not dimensionless, $\beta$ inverted, omits $\rho_0$/$\sin\gamma$; correct Allen–Eggers result is $h_q = H\ln[3\rho_0 H/(\beta\sin\gamma)]$. Peak heating is higher than peak deceleration (heating $\propto\sqrt\rho V^3$ vs $\rho V^2$). Flagged in Known Issues.