Atmospheric Escape Velocity Calculator — Planets

Whether a world keeps an atmosphere is a contest between gravity and molecular motion. Escape velocity follows from equating kinetic and gravitational potential energy, v_esc = sqrt(2GM/R), giving 11.19 km/s for Earth, 5.03 km/s for Mars, 2.38 km/s for the Moon and 59.5 km/s for Jupiter. Against that stands the root-mean-square thermal speed v = sqrt(3kT/m), which rises with temperature and falls with molecular mass, so at 288 K hydrogen molecules average about 1.9 km/s while nitrogen and oxygen average near 0.5 km/s. The decisive quantity is the ratio of the two. Because speeds follow a Maxwell-Boltzmann distribution with a thin high-speed tail, a gas survives billions of years only when v_esc exceeds the thermal speed roughly sixfold, is marginal between three and six, and is lost quickly below three. This explains the pattern across the solar system. Earth retains nitrogen, oxygen and carbon dioxide easily but has lost nearly all its free hydrogen and helium. Mars, with 38 percent of Earth surface gravity, holds a thin carbon dioxide atmosphere at about 610 Pa. Titan is colder than Mars at roughly 94 K, and that low temperature lets a body with a weaker pull than the Moon hold a nitrogen atmosphere denser than Earth. Mercury, hot and small, holds only a transient exosphere. The scale height kT/(m g) reported alongside describes how quickly pressure falls with altitude, about 8.5 km for Earth. The model covers only thermal Jeans escape, omitting hydrodynamic blowoff, solar wind stripping, photochemical escape and impact erosion, all of which shaped the Martian atmosphere.

Escape velocity is a direct consequence of the inverse-square gravitation Isaac Newton published in the Principia in 1687, though the phrase and its use in planetary science came much later. The kinetic theory needed for the other half of the comparison developed through the nineteenth century, when James Clerk Maxwell derived the distribution of molecular speeds in 1860 and Ludwig Boltzmann generalised it, establishing that a gas at any temperature contains a small fraction of molecules moving far faster than average. Together these results made the retention question quantitative for the first time. John James Waterston and later George Johnstone Stoney argued in the 1890s that the fastest molecules in the tail of the distribution should leak away from small bodies, and Stoney used the idea to explain why the Moon has no air and why Earth lacks free hydrogen. James Hopwood Jeans placed the calculation on a rigorous footing in the early twentieth century, deriving the escape flux from an exosphere in his 1916 book on the dynamical theory of gases; thermal escape is still called Jeans escape. The rough rule that escape speed must be about six times the thermal speed for retention over the age of the solar system was popularised in mid-twentieth-century planetary texts. Spacecraft since have shown thermal escape is only part of the story: NASA MAVEN, in orbit at Mars since 2014, measured ongoing loss of atmosphere to the solar wind.