Stratospheric Temperature⚠ unverified

Understanding the Parameters

• Altitude $h$ — geopotential altitude (see Geopotential Altitude). Only meaningful above $11{,}000\ \text{m}$; below that use ISA Temperature. As noted above, sub-tropopause values are not rejected — they silently return $228.65\ \text{K}$.

• The tropopause value $216.65\ \text{K}$ — this is not a new constant. It is exactly what ISA Temperature gives at 11 km: $288.15 - 0.0065 \times 11000 = 216.65\ \text{K}$. The branches are built to meet continuously, so the atmosphere has no temperature jumps.

• The isothermal layer (11–20 km) — for nine kilometres, standard temperature simply does not change. This is where airliners cruise, and it is why cruise temperature is roughly $-56\ ^\circ\text{C}$ whether at 35,000 or 39,000 ft — a convenience for engine and airframe design.

• The inversion (20–32 km): $+1\ \text{K/km}$ — the sign is the interesting physics. In the troposphere, air is heated from below (by the sunlit ground) and cools with height. In the stratosphere it is heated from above and within, by ozone absorbing solar ultraviolet. Temperature therefore climbs with altitude, and the rate accelerates higher up ($+2.8\ \text{K/km}$ above 32 km) as ozone heating intensifies, peaking near 47 km at the stratopause ($270.65\ \text{K}$ — nearly room temperature, at 47 km).

• Why the inversion means "stratified" — warm air sitting on top of cold air is statically stable: a displaced parcel is restored rather than accelerated. Vertical convection is suppressed, which is why the stratosphere is layered and smooth (no weather, no cumulus, minimal turbulence) and why the tropopause acts as a lid on the troposphere's storms. The layer's name is its physics.

• What it changes downstream — via ISA Pressure and ISA Density, temperature sets the whole state. The isothermal layer is exactly why ISA Pressure switches from a power law to a simple exponential above 11 km: constant $T$ makes the hydrostatic integral elementary.

Derivation (Approaching a Proof)

Like ISA Temperature, the stratospheric profile is defined, not derived — it is a piecewise-linear fit to observed mean soundings, adopted by standards bodies. There is no closed-form proof of $+1\ \text{K/km}$. What can be shown is why the sign flips, and why linear segments are the right descriptive language.

Why cooling must stop. The troposphere's lapse rate comes (see ISA Temperature) from adiabatic convection: $dT/dh = -g/c_p$. That derivation assumes air is vertically mixing. Convection requires a heat source at the bottom — the ground absorbing sunlight. But the atmosphere thins exponentially, and above ~11 km so little mass remains that surface-driven convection can no longer reach. Remove the mixing and the adiabatic argument, along with its lapse rate, simply does not apply. Temperature is then set by radiative balance, not convection.

Why it reverses. In radiative equilibrium a layer's temperature is set by absorbed versus emitted power. The stratosphere contains the ozone layer, peaking near 20–25 km. Ozone ($\text{O}_3$) absorbs strongly in the ultraviolet — the Hartley band near 250 nm — through the Chapman cycle:

$$\text{O}_2 + h\nu \to 2\text{O},\qquad \text{O} + \text{O}_2 + M \to \text{O}_3 + M,\qquad \text{O}_3 + h\nu \to \text{O}_2 + \text{O}.$$

Each cycle converts a UV photon's energy into kinetic energy of the surrounding molecules — that is, into heat. Crucially, the UV flux is strongest at the top (it is absorbed on the way down), while ozone density is greatest lower. The heating rate per unit mass is the product of flux and absorber density divided by air density, and because air density falls exponentially, the heating per kilogram increases with height. So:

$$\frac{dT}{dh} > 0 \quad\text{wherever ozone UV heating dominates.}$$

The temperature rises with altitude, and the rise steepens upward — precisely the observed structure ($0\ \text{K/km}$, then $+1$, then $+2.8$). The inversion is thus a direct signature of the ozone layer; without ozone, Earth would have no stratosphere in this sense.

Why linear segments. Radiative-photochemical equilibrium has no tidy analytic solution, so the standard does what standards do: fit straight lines to the observed mean profile and require continuity at the joins. Each layer is specified by a base altitude, a base temperature, and a lapse rate. Verifying continuity at the joins of this calculator's branches: at $h = 20\ \text{km}$ the second branch gives $216.65 + 0.001(20000-20000) = 216.65\ \text{K}$, matching the isothermal branch ✓; at $h = 32\ \text{km}$ it gives $216.65 + 0.001 \times 12000 = 228.65\ \text{K}$, matching the clamp ✓. The piecewise function is continuous — though its derivative is not, which is an artefact of the fit, not a real kink in the sky.

Dimensional check. $0.001\,(h - 20000) = (\text{K}/\text{m}) \times \text{m} = \text{K}$, which adds correctly to $216.65\ \text{K}$ ✓. The coefficient $0.001\ \text{K/m}$ is $+1\ \text{K/km}$, positive by construction — the sign carrying all the physics above.

History and Development

• Discovery (1902). Léon Teisserenc de Bort, launching hundreds of instrumented unmanned balloons from his private observatory at Trappes near Paris, found that above roughly 11 km the temperature stopped falling. The result was so contrary to expectation that he spent years checking for instrument error — solar heating of the thermometers was the obvious suspect — before announcing it in 1902. He named the layers troposphere and stratosphere. Richard Assmann in Germany reported the same finding independently and nearly simultaneously.

• Explaining it (1930). The discovery of the inversion long preceded its explanation. Sydney Chapman published the photochemical theory of the ozone layer in 1930 — the cycle sketched above now bears his name — identifying ozone's UV absorption as the stratosphere's heat source. Gordon Dobson, whose spectrophotometer and eponymous unit still measure ozone, mapped its global distribution through the 1920s–30s. Only then was the $+1\ \text{K/km}$ in this equation understood rather than merely tabulated.

• Standardisation (1952–1976). The ICAO Standard Atmosphere (1952) codified the troposphere and the isothermal layer; the 1964 extension to 32 km added the $+1\ \text{K/km}$ inversion layer implemented here, and the U.S. Standard Atmosphere 1976 carried the structure to 84.852 km, adding the $+2.8\ \text{K/km}$ layer (32–47 km) that this calculator clamps away, the stratopause at 47 km, and the mesosphere's renewed cooling above 51 km. ISO 2533:1975 agrees throughout.

• Why anyone cares now. The stratosphere stopped being an aeronautical curiosity when it became a policy battleground: the ozone layer's depletion by CFCs (Molina and Rowland, 1974; the Antarctic ozone hole, Farman et al., 1985) and the Montreal Protocol (1987) all turn on the chemistry that puts the $+$ sign in this equation. Cooling of the stratosphere is also a fingerprint of greenhouse warming below.

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

Notes: ⚠ Default input $h = 1.0\ \text{m}$ is broken — it misses every branch and returns the $228.65\ \text{K}$ clamp (the 32 km value) for sea level; there is no domain guard for $h < 11{,}000\ \text{m}$. Name/equation mismatch: the shown formula is the 20–32 km layer, while the lower stratosphere (11–20 km) is the isothermal $216.65\ \text{K}$ branch. Equation box shows 1 of 3 branches (isothermal 11–20 km; $+1\ \text{K/km}$ 20–32 km; clamped 228.65 above — the real ISA continues at $+2.8\ \text{K/km}$ to $270.65\ \text{K}$ at the 47 km stratopause). $216.65\ \text{K}$ is ISA Temperature evaluated at 11 km — branches meet continuously. The temperature inversion is the ozone layer's signature (Chapman cycle UV absorption), and it is what makes the stratosphere stably stratified.