🌍 Planet Atmospheres

Scale height H is the vertical distance over which atmospheric pressure falls by a factor of e (about 37 per cent). It equals k_B·T divided by the mean molecular mass times gravity, so warmer, lighter, low-gravity atmospheres are puffier. That is why cold but low-gravity Titan has H near 21 km while Earth’s is about 8.5 km.

It first finds the radiative equilibrium temperature from the absorbed sunlight, T_eq = (S(1−α)/4σ)^¼, where S is the solar flux scaled by distance and luminosity. It then adds a greenhouse increment that grows logarithmically with CO₂, calibrated to give Earth its +33 K boost. The surface temperature shown is T_eq plus that increment.

λ compares a gas molecule’s gravitational binding energy to its thermal energy: λ = v_esc²·m/(2k_B·T). A large λ (roughly above 15–30) means the atmosphere is well retained, while a small value means light molecules can readily escape to space. This is a key reason low-gravity Mars and the Moon struggle to hold onto their gas.

Mars has a low escape velocity of about 5 km/s and lacks a strong global magnetic field, so over billions of years thermal escape and stripping by the solar wind removed most of its air. Today its surface pressure is only about 0.006 atmospheres, around 0.6 per cent of Earth’s, which the pressure profile makes visible at a glance.

The framework is correct in form — the barometric law, radiative equilibrium, Stefan-Boltzmann emission and the Jeans parameter are all real physics. The numbers are simplified, however: scale height is treated as constant with altitude, the greenhouse term is a single calibrated logarithmic fit rather than a full radiative-transfer calculation, and clouds are reduced to one albedo value. It is an educational comparison, not a research climate model.