ISA Temperature✓ verified

Understanding the Parameters

• Altitude $h$ — height above the sea-level datum, in metres. The formula is a straight line in $h$, so every kilometre of climb costs exactly $6.5\ \text{K}$: $5\ \text{km} \to 255.65\ \text{K}$ ($-17.5\ ^\circ\text{C}$), and $11\ \text{km} \to 216.65\ \text{K}$ ($-56.5\ ^\circ\text{C}$), the coldest point of the standard troposphere.

• Sea-level datum $T_0 = 288.15\ \text{K}$ — exactly $15\ ^\circ\text{C}$. A round, deliberately chosen number approximating the global mean surface temperature at mid-latitudes. It is a definition, not a measurement.

• Lapse rate $L = 0.0065\ \text{K/m}$ — the rate at which standard temperature drops with height ($6.5\ \text{K/km}$). This is the physically important constant: it is close to, but deliberately below, the dry adiabatic lapse rate of $\approx 9.8\ \text{K/km}$, because real air carries moisture whose condensation releases latent heat and softens the decline. It is an observed average, not a derived constant.

• The 11 km ceiling — the linear fall stops at the tropopause. Above it lies the isothermal lower stratosphere, where temperature holds at $216.65\ \text{K}$ up to 20 km before rising again. This calculator clamps at the tropopause value and never rises.

• ISA deviation — real air is rarely at ISA. Pilots and engineers quote conditions as "ISA $+15$" meaning $15\ \text{K}$ hotter than standard at that altitude. Hot days mean thinner air, longer takeoff runs, and less thrust — the practical reason the standard exists at all (see Density Altitude).

Derivation (Approaching a Proof)

Unlike ISA Pressure and ISA Density, which follow from physics, the temperature profile is the defining assumption of the ISA — the model's axiom, from which everything else is derived. Its form is chosen, not proven. But it is not arbitrary; here is why a linear profile with this slope is the right choice.

Consider a parcel of dry air rising slowly enough to stay in pressure equilibrium with its surroundings, but fast enough to exchange no heat (adiabatic). The first law for a reversible adiabatic process gives

$$c_p\,dT = \frac{1}{\rho}\,dP,$$

and hydrostatic balance (see ISA Pressure) gives $dP = -\rho g\,dh$. Substituting,

$$c_p\,dT = -g\,dh \quad\Longrightarrow\quad \frac{dT}{dh} = -\frac{g}{c_p}.$$

With $g = 9.80665\ \text{m/s}^2$ and $c_p \approx 1005\ \text{J}/(\text{kg}\cdot\text{K})$ for dry air, this dry adiabatic lapse rate is

$$\Gamma_d = \frac{9.80665}{1005} \approx 0.00976\ \text{K/m} \approx 9.8\ \text{K/km}.$$

So the physics predicts a constant slope — the linear form of $T(h)$ is exactly what vertical mixing of a well-stirred atmosphere produces. The magnitude, however, is too steep: the real troposphere averages $6.5\ \text{K/km}$, not $9.8$. The gap is moisture. Rising air cools, water vapour condenses, and the released latent heat partially offsets the cooling, giving a saturated lapse rate as low as $4\text{–}5\ \text{K/km}$ in warm humid air. The ISA's $6.5\ \text{K/km}$ is the observed global average of a real, partly-moist troposphere — sitting between the dry and saturated limits, exactly where it should.

Integrating the constant slope from the sea-level datum:

$$\int_{T_0}^{T} dT' = \int_0^h (-L)\,dh' \quad\Longrightarrow\quad T - T_0 = -L h \quad\Longrightarrow\quad T = T_0 - L h.$$

The linear profile is thus derived in form from adiabatic convection plus hydrostatic balance, and calibrated in magnitude by observation.

Dimensional check. $L h = (\text{K}/\text{m}) \times \text{m} = \text{K}$, which subtracts correctly from $T_0$ in K to give $T$ in K.

History and Development

• Early stratification (1900s). Léon Teisserenc de Bort, flying unmanned instrumented balloons from Trappes near Paris through the late 1890s, found that temperature stopped falling above roughly 11 km. He announced the result in 1902 and named the two layers the troposphere ("turning sphere", where mixing occurs) and the stratosphere ("layered sphere"). The 11 km break in this calculator is his discovery.

• First standards (1920s). Aviation's growth demanded a common reference for comparing aircraft and calibrating altimeters. The US and several European nations adopted national standard atmospheres in the 1920s; ICAN (the International Commission for Air Navigation) issued an early international table.

• ICAO Standard Atmosphere (1952). The International Civil Aviation Organization adopted the standard that fixed $T_0 = 288.15\ \text{K}$, $P_0 = 101{,}325\ \text{Pa}$, and $L = 6.5\ \text{K/km}$ — the values used here. It was extended to 32 km in 1964 and to 80 km in 1976.

• Convergence (1976). The U.S. Standard Atmosphere 1976 and ISO 2533:1975 align with ICAO through the lower layers, so "ISA", "ICAO Standard Atmosphere", and "US Standard Atmosphere" agree below 32 km. The choice of a defined rather than measured profile is the whole point: two engineers on different continents in different weather compute the same number.

Related Concepts: ISA Pressure, ISA Density, Stratospheric Temperature, Geopotential Altitude, Speed Of Sound, Density Altitude, Pressure Altitude

Notes: $T_0$ and $L$ are hard-coded constants, not inputs. Input labelled "geometric" but ISA is defined on geopotential altitude — see Geopotential Altitude. Displayed equation is the troposphere branch only; the implementation clamps to $216.65\ \text{K}$ above 11 km (correct to 20 km, wrong above — see Stratospheric Temperature). The ISA is a defined standard, not a forecast.