The Moon Has No Atmosphere

Unlike the Earth, the Moon has essentially no atmosphere. The Moon might have had an atmosphere when it was formed, or outgassed significant atmosphere since then, but it has since lost any previous atmosphere. Thus, it has no protective blanket to moderate its temperatures (see the right panel) or to shield it from meteors.
Retention of Atmospheres
To understand why the Earth has an atmosphere but the Moon does not, we need to consider four basic factors governing whether a planet or a moon retains an atmosphere: (1) the mass of the planet; (2) the mass of the gas molecules in the atmosphere; (3) the temperature of the surface and atmosphere; (4) the strength of the magnetic field.

The first factor governs how strongly the planet attracts the gas in its atmosphere, the second and third govern how rapidly the molecules move in the atmosphere, and the fourth determines whether there is a magnetosphere to shield the atmosphere from the solar wind.

Maxwell Velocity Distribution
The atoms and molecules in a planetary atmosphere have a distribution of velocities called a Maxwell distribution (see adjacent figure). This graph plots the number of atoms or molecules with a given velocity versus the velocity. The higher the temperature, the higher the average velocity (indicated by the dotted vertical lines) and the higher the velocity of the fastest molecules. In addition, the higher the mass of a molecule at a given temperature, the lower the average velocity in the atmosphere. For example, at a fixed temperature, light hydrogen molecules will on average be moving faster than heavy oxygen molecules.

Escape Velocity
 The critical question is whether the fastest molecules attain escape velocity, which is determined by the mass of the planet (here is an escape velocity calculator for planets in the Solar System). High temperatures, light gases, and weak gravitational fields favor the loss of the atmosphere into space.

The situation in the Solar System is illustrated in the adjacent diagram. The yellow dots indicate the escape velocities of various objects in the Solar System, plotted versus the temperature at the tops of their atmospheres (or their surfaces if they have no atmosphere). The green curves indicate the typical highest velocities for different gas molecules as a function of temperature. Generally, if the point representing the escape velocity for an object is well above the curve for a given atom or molecule, it is likely that the object can retain the atom or molecule in its gravitational field. The rough "rule of thumb" is that the escape velocity needs to be larger by a factor of about 6 for the gas to have been retained by the planet or moon.

Magnetospheres and the Solar Wind

The solar wind of charged particles from the Sun can interact with atmospheres and strip ions from the upper part if there is no magnetic field to deflect the wind. The Moon has no magnetic field, but this is not important because it has lost any atmosphere that it had anyway. However, Venus and Mars also lack a strong magnetic field. Scientists believe that their inability to shield themselves from the solar wind has had a significant influence on the evolution of their atmospheres.

Atmospheres in the Solar System
So, for example, the gas giant planets have sufficiently high escape velocities that they can retain with high probability all the gases listed. Since the majority of the original solar nebula was hydrogen and helium, they have large concentrations of these gases. At the other extreme, low-mass objects like the Moon or the asteroid Ceres cannot retain any of the gases listed.

Planets like the Earth are intermediate cases: the Earth easily retains heavy gases like oxygen or water vapor, but has lost most of its free hydrogen and helium (there is significant hydrogen on Earth in water, but water is a heavy enough molecule to be retained by the gravitational field).

The Moon Has Insufficient Gravity to Retain an Atmosphere
We conclude that the Moon has no atmosphere because its gravitational field is too weak to have retained any atmosphere that it had over times comparable to the age of the Solar System. Notice that retention of an atmosphere depends on the temperature. The Saturnian moon Titan has an escape velocity only about twice that of the Moon, but it has retained an atmosphere even thicker than that of the Earth because it is far from the Sun and therefore much colder than the Moon.