Barometric Altitude⚠ unverified

Understanding the Parameters

• How every altimeter works — an altimeter is a barometer with a height scale. Air pressure falls predictably with height (see ISA Pressure); measure the pressure and invert that relation to read off an altitude. There is no direct "height" sensor in a basic altimeter — only pressure, converted by this formula.

• The reference pressure $P_0$ is the altimeter setting — the crucial operational input. Setting $P_0$ to the local sea-level pressure (QNH) makes the altimeter read height above sea level; setting it to the standard $1013.25\ \text{hPa}$ (QNE) makes it read pressure altitude / flight level, the datum all high-altitude traffic shares. Dialing $P_0$ is literally turning the "Kollsman window" knob in the cockpit.

• Why it needs a temperature reference $T_0$ — the pressure-height relation depends on the temperature profile of the air column (warm air is less dense, so pressure falls more slowly with height). The ISA assumes a standard $T_0$ and lapse rate; on a non-standard day the true height differs from the indicated barometric height — the origin of cold-weather altimeter corrections ("high to low, look out below").

• Only valid in the troposphere — the formula uses the linear-lapse troposphere branch of the ISA, so it is accurate to about $11\ \text{km}$. Above the tropopause the atmosphere goes isothermal and pressure decays exponentially instead (see ISA Pressure, Stratospheric Temperature); this formula would need its stratospheric branch there.

• Relationship to the Atmosphere calculators — this is the inverse of ISA Pressure (which gives $P$ from $h$) and is essentially Pressure Altitude with $P_0$, $T_0$ user-settable. For performance you also want Density Altitude (which additionally uses temperature to get air density).

Derivation (Approaching a Proof)

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

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

where $L$ is the lapse rate and $g/(LR)$ the ISA pressure exponent ($\approx 5.2559$). Solve for $h$. Divide by $P_0$ and raise both sides to the reciprocal power $LR/g$:

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

Rearrange for $h$:

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

Insert the constants used by the implementation: $T_0/L = 288.15/0.0065 = 44{,}330.8\ \text{m}$ and $LR/g = 0.0065\times287.05/9.81 = 0.19020$, giving the shipped form $h = 44{,}330.8\,T_0/288.15\cdot(1-(P/P_0)^{0.19020})$ (with $T_0/L$ scaling if $T_0$ is changed). This is identical in structure to Pressure Altitude — the same inversion of the same law — with $P_0$ and $T_0$ promoted from hard-coded constants to inputs.

Sanity check. At $P = P_0$ the bracket is $1-1=0$, so $h=0$ (you are at the reference level). As $P\to0$, $h\to T_0/L = 44.3\ \text{km}$ — the (fictitious) altitude at which the extrapolated linear-lapse atmosphere would reach zero pressure, a mathematical artefact of the troposphere model, not a real ceiling.

Dimensional check. The exponent $LR/g$ is dimensionless (it is $1/5.2559$; see ISA Pressure for the unit cancellation), so $(P/P_0)^{LR/g}$ is dimensionless and $h$ carries the metres of $T_0/L$: $\text{K}/(\text{K}/\text{m}) = \text{m}$ ✓.

History and Development

• Pressure as a height proxy (1648). Blaise Pascal's Puy-de-Dôme experiment established that atmospheric pressure falls with altitude, making a barometer a height sensor — the founding idea of all barometric altimetry (see Pressure Altitude for the fuller history).

• The sensitive altimeter (1928). Paul Kollsman's precision aneroid altimeter with an adjustable barometric sub-scale (the "Kollsman window", the physical $P_0$ input here) made instrument flight practical and standardised how pilots set the reference pressure.

• Standard datum and flight levels. Because barometric altitude is only as good as its reference, aviation adopted the convention that above a transition altitude everyone sets the standard $1013.25\ \text{hPa}$ and flies flight levels — guaranteeing that all aircraft share one pressure datum even if it disagrees with true height. This calculator, with $P_0$ exposed, is exactly the computation an altimeter performs once the pilot has dialed in the setting.

Related Concepts: Pressure Altitude, ISA Pressure, ISA Temperature, Density Altitude, Geopotential Altitude, Stratospheric Temperature

Notes: Near-duplicate of Pressure Altitude with $P_0$, $T_0$ exposed as inputs (the altimeter setting / QNH). Same ISA troposphere inverse: $T_0/L = 44{,}330.8$ m, exponent $LR/g = 0.19020$ (uses $g=9.81$ vs Pressure Altitude's $9.80665$). Set $P_0 = 1013.25$ hPa for flight levels, local QNH for height AMSL. Troposphere only (~11 km); above, the atmosphere is isothermal (ISA Pressure). Default $P = 1\ \text{Pa}$ is a placeholder.