The fastest particles get the first chance
Imagine a gas particle near the top of an atmosphere. A collision sends it outward. Gravity slows it; most particles turn around and return. A sufficiently fast particle can keep going, provided that another collision does not redirect it first.
Temperature describes a distribution of random speeds. Even a cold gas has a small fast tail. Raising the temperature adds particles to that tail; increasing particle mass or the planet’s gravity makes escape harder. The usual comparison is between gravitational binding energy and thermal energy per particle.
Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1
Why the top of the atmosphere matters
Deep down, particles collide before they travel far. An unusually fast particle generally shares its energy with its neighbors. Higher up, the gas is thinner and the distance between collisions grows. The outer region where that distance becomes comparable to the distance over which density changes is called the exobase.
At a strongly bound exobase, the gas below can be nearly stationary and the outward fast tail can provide a small leak. This calculation is called Jeans escape. Counting fast particles is not enough: the particles moving most quickly outward cross a surface more often, so the escape calculation weights them by outward velocity. The following chapter builds that integral.
Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1
The atmosphere can also move together
Now warm the gas enough that pressure can drive a sustained outward flow. Collisions transmit that push between neighbors. The gas near the bottom may drift slowly even while the outer atmosphere streams away rapidly.
A useful first model keeps temperature fixed at every radius. Its accelerating solution passes through the sound speed at a particular radius. Below that point, pressure disturbances can travel upstream through the subsonic flow; above it, the outward flow outruns them. This is the isothermal Parker wind, first developed for stellar outflows and useful as a deliberately simplified planetary transport model.
The amount of gas carried by this solution still needs a density at its lower boundary. Two atmospheres at the same temperature and gravity can have the same velocity shape but carry different amounts of mass. The shape of a wind and the supply of gas are separate pieces of the calculation.
Reading: Cranmer 2004, New views of the solar wind with the Lambert W function, v1
A hot wind needs a heat source
Gas gains gravitational potential energy as it climbs away and does work as it expands. Keeping it warm requires a continuing supply of heat. A star’s X rays and extreme ultraviolet radiation can supply some of that energy. Only part of the intercepted power becomes mechanical escape; ionization and radiation can take substantial shares.
A quick energy-limited estimate divides usable incoming power by the binding energy per kilogram. It gives a useful scale. A complete atmosphere calculation also follows where the light is absorbed, how the gas cools and how temperature changes along the flow.
Small planets can additionally lose gas while their interiors and envelopes cool. Their changing radius alters the atmosphere through which radiation travels. The observed shortage of intermediate-size close-in planets is called the radius valley. Connecting those processes to it takes us from an individual wind to the history of a population. The last guide follows that argument through fixed primary papers.
Reading: Murray-Clay, Chiang & Murray2009, Atmospheric escape from hot Jupiters, v2 · Ginzburg, Schlichting & Sari 2018, Core-powered mass loss, v2 · Owen & Wu 2017, The evaporation valley in the Kepler planets, v2