How does escape velocity explain that many moons don't have atmospheres?

Answer 1

Some fraction of the molecules in the atmosphere exceed escape velocity. Eventually, all are effectively lost, as a fraction of the remainder will also exceed escape velocity, ad infinitum.

Answer 2

The gas molecules have a statistic distribution of speed. The probable speed is a function of the molecular mass and the temperature.
If a molecule is locate in the space border (that's mean a low probability to collide with more molecules) then it have some probability to have a higher speed than the escape speed for the moon or planet. The smaller the molecule, smaller the gravity, the higher the temperature, the easier the molecule can escape to the space. Few moons have an atmosphere. Titan is one of the largest moon, and have an atmosphere. Ganymede is little larger but little warmer too, so it has less atmosphere
Venus gravity is less than Earth's and the surface temperature is high, but the gases are heavier (sulfuric acid) than Earth's so the Venus atmosphere is thicker than Earth's.

As some gases can come from the Sun, from the comets or from the volcanoes, the planets and moons may increase or loose part of the atmosphere upon the time, in a slow or fast way.

Answer 3

To keep an atmosphere, the air molecules have to moving at less than the escape velocity for that world.
The smaller the mass, the lesser the gravity and therefore the lower the escape velocity. And the easier it is for even slow-moving molecules to escape to space.

Answer 4

In the kinetic theory of gasses, the molecules have a velocity dependent on temperature.
When a molecule has escape velocity, ...it's gone.
Moons generally have low gravity and thus low escape velocities.
Only larger moons are observed to retain an atmosphere.
Smaller moons, (excepting those generating gas by vulcanism), generally are not.

Answer 5

Gravity

Answer 6

Generally, the lower the gravity field, the more likely it is to be without an atmosphere.