Pressure Altitude✓ verified

Understanding the Parameters

• Static pressure $P$ — the measured ambient pressure, sensed through a static port sited where the airflow neither compresses nor rarefies the air. This is the only input: pressure altitude deliberately ignores temperature. The default $90{,}000\ \text{Pa}$ gives $h_p \approx 988\ \text{m}$.

• Pressure altitude $h_p$ — an instrument reading, not a height. If the day is colder or warmer than standard, the real geometric height differs from $h_p$, sometimes by hundreds of metres. Pressure altitude is a label for a pressure surface, and admitting that is the key to understanding altimetry.

• The constant $44{,}330.8\ \text{m} = T_0/L = 288.15/0.0065$ — where the ISA's straight-line temperature profile would hit $0\ \text{K}$ if extended forever. It is a mathematical artefact of the linear fit, not a physical altitude; nothing happens there. It appears because inverting the power law naturally produces this scale factor. The same constant appears in Density Altitude for exactly the same reason.

• The exponent $0.190263 = 1/5.2559$ — the reciprocal of $g/(LR)$. Whenever you invert a power law, the exponent inverts; this number carries no independent meaning.

• Why separation uses pressure, not truth — this is the crucial insight. Two aircraft with correctly set altimeters in the same air mass are on the same pressure surface when they read the same altitude, and therefore genuinely separated — even if both readings are wrong by 300 m. A shared error is harmless; an unshared correction is dangerous. Above the transition altitude everyone sets 1013.25 hPa and flies flight levels (FL350 = pressure altitude 35,000 ft), guaranteeing consistency at the price of accuracy.

• The cold-weather trap — "high to low, look out below". Flying from high pressure to low, or warm air to cold, the altimeter over-reads: it says you are higher than you are. In cold air the pressure surfaces bunch lower, so a true-height terrain clearance evaporates. This is not a defect of the instrument but a direct consequence of assuming ISA temperature when the day disagrees — which is why cold-temperature altitude corrections are mandatory on approach procedures in winter.

Derivation (Approaching a Proof)

Begin with the troposphere pressure law derived in full from hydrostatic balance and the ideal gas law in ISA Pressure:

$$P = P_0\left(\frac{T}{T_0}\right)^{g/(LR)}, \qquad T = T_0 - L h.$$

Substituting the temperature profile expresses pressure directly in altitude:

$$P = P_0\left(\frac{T_0 - Lh}{T_0}\right)^{g/(LR)} = P_0\left(1 - \frac{Lh}{T_0}\right)^{g/(LR)}.$$

Now solve for $h$. Divide by $P_0$:

$$\frac{P}{P_0} = \left(1 - \frac{Lh}{T_0}\right)^{g/(LR)}.$$

Raise both sides to the reciprocal power $LR/g$ to undo the exponent:

$$\left(\frac{P}{P_0}\right)^{LR/g} = 1 - \frac{Lh}{T_0}.$$

Rearrange for $h$:

$$\frac{Lh}{T_0} = 1 - \left(\frac{P}{P_0}\right)^{LR/g} \quad\Longrightarrow\quad h = \frac{T_0}{L}\left(1 - \left(\frac{P}{P_0}\right)^{LR/g}\right).$$

Insert the standard constants. The scale factor is

$$\frac{T_0}{L} = \frac{288.15}{0.0065} = 44{,}330.77\ \text{m},$$

and the exponent is

$$\frac{LR}{g} = \frac{1}{g/(LR)} = \frac{1}{5.2559} = 0.190261,$$

giving the implemented formula (which rounds these to $44{,}330.8$ and $0.190263$):

$$h_p = 44{,}330.8\left(1 - \left(\frac{P}{101325}\right)^{0.190263}\right). \qquad\blacksquare$$

Sanity checks. At $P = P_0 = 101{,}325\ \text{Pa}$ the bracket is $1 - 1 = 0$, so $h_p = 0$ ✓. At $P = 22{,}632\ \text{Pa}$ (the tropopause), $(22632/101325)^{0.190263} = 0.7519$, giving $h_p = 44{,}330.8 \times 0.2481 \approx 11{,}000\ \text{m}$ ✓ — the inversion recovers the tropopause exactly, confirming it is a true inverse of ISA Pressure within the troposphere. Above sea-level pressure ($P > P_0$, as in a deep low-pressure system at sea level, or below sea level) the bracket goes negative and $h_p < 0$ — correctly reporting a negative pressure altitude, which is physical and routine.

Dimensional check. $P/P_0$ is a dimensionless pressure ratio; raising it to any power leaves it dimensionless, so the bracket is dimensionless and $h_p$ carries the units of $T_0/L$: $\text{K}/(\text{K}/\text{m}) = \text{m}$ ✓.

History and Development

• Height from pressure (1648). Blaise Pascal's Puy de Dôme experiment — carrying a barometer up an Auvergne volcano and watching the mercury fall — established that pressure could serve as a proxy for height. Every altimeter since is a footnote to that afternoon.

• The hypsometric formula (1800s). Pierre-Simon Laplace formalised barometric levelling; 19th-century surveyors and mountaineers routinely determined elevations barometrically, wrestling with exactly the temperature-dependence that modern altimetry sidesteps by defining it away.

• The aneroid altimeter (1920s–30s). Paul Kollsman's sensitive aneroid altimeter (1928) — with an adjustable barometric sub-scale, the "Kollsman window" — made instrument flight practical. Jimmy Doolittle's September 1929 "blind flight" at Mitchel Field, flown entirely on instruments under a hooded cockpit, depended on it. The sub-scale is the physical admission that pressure altitude is a convention: the pilot dials in which convention they are using.

• The flight-level system. As traffic densified, the realisation that consistency beats accuracy was formalised: above a transition altitude, all aircraft set the standard datum 1013.25 hPa (29.92 inHg) and read flight levels off this very equation. Everyone is wrong together, and therefore safe. The QNH/QFE/QNE Q-code settings encode which datum is in use.

• Standardisation (1952– ). The ICAO Standard Atmosphere fixed the constants, making this inversion the same worldwide. Modern air data computers evaluate it digitally, but the equation is unchanged — and modern GNSS gives true geometric height directly, which is precisely why it cannot replace pressure altitude for separation without changing the rules of the game.

Related Concepts: ISA Pressure, ISA Temperature, ISA Density, Density Altitude, Geopotential Altitude, Stratospheric Temperature, Dynamic Pressure, Bernoulli Total Pressure

Notes: Registry latex_equation is the opaque placeholder $h_p = f(P)$ — the real formula is the inverse troposphere power law above; should be corrected. $44{,}330.8\ \text{m} = T_0/L$ (where the linear ISA temperature profile would reach 0 K — a mathematical artefact, not a physical height); exponent $0.190263 = 1/5.2559$. Troposphere inverse only — above 11 km it extrapolates rather than inverting ISA Pressure's isothermal branch, which is correct by definition for altimetry but means it is not this app's own ISA inverse up there. Pressure altitude is an instrument convention, not a true height; separation relies on shared error.