Why the air thins with height and your ears pop: gravity pulls the atmosphere down, pressure holds it up, and the balance makes pressure fall exponentially — not linearly — halving every few kilometres. Rendered, not quoted.
source Pascal, Récit de la grande expérience de l'équilibre des liqueurs (Puy-de-Dôme experiment carried out by F. Périer, 19 Sept 1648); formula formalized by Laplace, Traité de Mécanique Céleste vol. 4 (1805). No stable primary DOI — cited author/title/year, marked AMBER. context
Hydrostatic balance: a slab of air of thickness dz is held up by the pressure difference across it against its own weight.
For an isothermal column this integrates to a pure exponential:
H is the scale height — the rise over which pressure drops by a factor of e (≈2.718). For Earth's dry air near 273 K, H ≈ 8.0 km.
Air thins with height — the barometric formula: P = P₀·e^(−z/H), H = RT/Mg ≈ 8 km, halving every ≈5.5 km. The same hydrostatic balance that sets this pressure structure sits directly above the-adiabatic-lapse-rate: pressure falls exponentially, and a parcel dragged up that pressure gradient cools at g/c_p ≈ 9.8 K/km.
Barometer → lapse rate → the whole vertical thermodynamics of the troposphere.
Re-runs the pure engine on load and confirms the exponential holds, P>0 everywhere, and the scale-height dependencies. Flips red if the machine is tampered.
Surface pressure P₀ = 101 325 Pa, temperature T (K), molar mass M (kg/mol), gravity g = 9.80665, gas constant R = 8.31446.
Drag the temperature to watch the column swell or shrink.
Live: pressure profile P(z) from surface to 30 km. The dashed line marks one scale height H (P falls to 1/e); the dotted line marks the half-height H·ln2.
Proven: pressure and density fall on the same exponential, never reaching zero — the atmosphere has no top edge, it just fades.
True, and named honestly: the troposphere cools ~6.5 K/km, so a single H under-shoots high up and the exact profile needs T(z). But the form survives — locally P still obeys dP/dz=−ρg, and the isothermal law is the exact solution of the clean limit and a tight approximation over a scale height. The exponential is the skeleton, not the fiction.
Pressure falls off linearly and the air ends at a definite ceiling. → Exponential; P>0 for all finite z — no ceiling, only thinning.
Heavier air (more humid) presses harder, so humid air is denser. → Water vapour (M≈18) is LIGHTER than dry air (M≈29); humid air is less dense — larger effective H.
Halving height and 1/e height are the same distance. → Half at H·ln2 ≈ 5.5 km; 1/e at H ≈ 8.0 km. ln2 < 1.
Swap the engine to the linear model P = P₀·(1 − z/H) — pressure hits zero at z=H and goes negative above. The Witness (7) catches it live.