The Exponential Atmosphere
Planetary atmospheres thin exponentially with altitude — pressure at height z equals surface pressure times exp(-z/H), where H is the scale height. This single parameter captures the interplay of gravity pulling molecules down and thermal energy pushing them apart. A warm, low-gravity world with light gases (like a hot Jupiter) can have scale heights of hundreds of kilometers, while a cold, massive world with heavy molecules packs its atmosphere into a thin shell.
Escape Velocity and Molecular Speed
Every gas molecule moves with a speed drawn from the Maxwell-Boltzmann distribution. The RMS thermal velocity v_th = sqrt(3kT/m) sets the typical speed. If the planet's escape velocity v_esc = sqrt(2GM/R) is not much larger, molecules in the high-speed tail of the distribution continuously leak away into space — a process called Jeans escape that operates over millions to billions of years.
The Retention Criterion
Planetary scientists use the ratio v_esc/v_th as a quick diagnostic. Ratios above 6 indicate stable retention over Solar System age; below 4, the gas is lost geologically fast. Earth easily holds N₂ and O₂ (ratio ~22) but has lost most of its primordial hydrogen (ratio ~3.5). This explains why terrestrial planets have secondary atmospheres from volcanic outgassing while gas giants keep their primordial hydrogen envelopes.
Applications to Exoplanets
The same physics governs exoplanet atmospheres. Hot Jupiters orbiting close to their stars have such high temperatures that even hydrogen approaches the escape threshold, producing dramatic atmospheric mass loss observed as extended hydrogen exospheres. NASA's James Webb Space Telescope measures transmission spectra that directly probe atmospheric scale heights, connecting theory to observation.