Review of the Structure of the Outer Atmosphere
      In the upper reaches of a planet's atmosphere (the exosphere and mesosphere) ultraviolet radiation from the Sun easily penetrates the thin (low-density) gases, is absorbed by the atoms and molecules which make up these gases, and in the process breaks them down into smaller pieces. Many of these pieces are electrically charged (any broken-up atoms, and many molecular fragments), so the gas at high altitudes consists to a great extent of ions, or charged particles. The region thus affected is referred to as the ionosphere. The ionosphere is not a separate part of the atmosphere from the regions which are defined according to their density and temperature; it overlaps them and extends from the top of the atmosphere to however deeply the Sun's ultraviolet radiation can penetrate into the atmosphere.
      During the day, when the Sun is up, the ionosphere gradually extends deeper and deeper into the atmosphere, as the Sun's ultraviolet radiation breaks down more and more molecules (this process is referred to as photo-dissociation if molecules are being broken down, and photo-ionization if atoms are being broken down). If it were not for an opposing process, recombination (building up pieces of atoms or molecules into the original or new structures), the ionosphere might reach the surface of the planet. However, as the radiation penetrates into deeper denser layers, collisions between the particles become more and more frequent (the rate of collision increases approximately as the square of the density: if there are more particles there are more collisions both because there are more particles to collide, and because there are more particles for them to collide with). As the ionosphere reaches deeper into the atmosphere the denser gases recombine more and more quickly, and at some point recombine just as fast as they are being broken down. The ultraviolet radiation which is capable of breaking down the gas particles cannot penetrate any further than this point, and the ionosphere cannot come any closer to the surface of the planet.
      Close to the surface of the planet temperature decreases with altitude, as the primary heat source is the absorption of light by the planetary surface, the contact of the lower atmosphere with that surface, and to a certain extent vertical mixing or convection, which occurs when the temperature gradient becomes too large for the atmosphere to remain stable against convection. In this region the atmosphere has a relatively constant composition, as occasional storms, accompanied by convective motions, thoroughly mix the atmosphere in relatively short times. But in the upper atmosphere (particularly the exosphere) the temperature is either relatively constant or even increases with height. This increase in temperature is caused by two factors: the absorption of ultraviolet light and the energy of the solar wind.
      For planets that have no significant magnetic field such as Venus and Mars the solar wind can run directly into the outer atmosphere of the planet. The solar wind is very rarefied (consisting of only a few atoms to a few hundred atoms per cubic inch, versus billions of trillions of atoms per cubic inch in the gas near the surface of the planet), but it is moving outwards from the Sun at tremendous speeds (200 to 300 miles per second when the 'weather' on the Sun is relatively quiet, and two to three times that fast when the Solar weather is relatively violent). As a result, the solar wind contains a huge amount of energy. This energy, when delivered to the thin gases at the top of a planet's atmosphere, can either strip the gas away from the planet or give it large kinetic energies (energy of motion), which correspond to very high temperatures. If you were to place a thermometer in the thin gases at the top of the atmosphere, it would not register any heat due to the gas, because there is essentially nothing there; but the high speeds of the gas particles would correspond to a temperature of as much as several thousand degrees if the gas were thick enough to affect the thermometer. Under these circumstances we say that the gas has a kinetic temperature of several thousand degrees, even though its total heat content is very small (because of these high temperatures the outer exosphere is sometimes called the thermosphere).
      In the outer reaches of the atmosphere there is practically no gas, and the energy deposited into the gas by the solar wind and by the absorption of ultraviolet radiation can heat the gas to thousands of degrees. But as you go down into the atmosphere the amount of heating gradually diminishes, partly because some of the energy that is being dumped into the atmosphere has already been absorbed at higher altitudes, and partly because the gas is getting denser, which means that the energy is spread among more particles, leaving less energy per particle. When discussing the kinetic temperature of a gas it is the energy per particle that determines how hot the gas is, so if there are more particles sharing the energy, the kinetic temperature is lower. For this reason, as you go down into the exosphere the deeper you get and the denser the gas gets, the lower the temperature is. The low-temperature boundary between the exosphere and the mesosphere (which is called the mesopause) represents the depth at which the energy of the solar wind has been completely absorbed by the upper atmosphere.

Motions of Particles in a Gas: The Maxwell-Boltzmann distribution
      The kinetic temperature of a gas depends upon and conversely determines the speed of the particles which make up the gas. If the temperature is low the particles move slowly (in fact if they were not moving at all we would say that the gas has a temperature equal to absolute zero); if the temperature is high the particles move more quickly. It is the relationship between motion and temperature that causes us to define the "kinetic" (meaning, related to motion) temperature of the gas. If the gas is thick enough to measure its temperature with a thermometer, the kinetic temperature and the thermometric temperature are the same; but if the gas is too thin to use a thermometer, the kinetic temperature is still defined by the energy of motion of the gas particles.
      At the surface of the Earth the average speed of the molecules (primarily nitrogen and oxygen) which make up the atmosphere is about 1/5 of a mile (1/3 of a kilometer) per second, or a little less than 700 miles (or a little more than 1000 kilometers) per hour. This average speed determines the rate at which sound waves, which represent a vibrational disturbance in the normally random motions of the particles, can spread out, so these numbers also represent the speed of sound at the surface of the Earth.
      Not all of the molecules actually move at this speed. As the molecules bounce off of each other they can change speeds in an erratic, "random" way. Often a collision will result in one molecule gaining energy at the expense of another one; the one which gains energy will go faster, and the one which loses energy will go slower. Sometimes both particles can 'lose' energy. As an example, suppose that they were headed towards each other in opposite directions; when they collide they might nearly come to a stop, which means that they would both suffer a reduction in their energy of motion (of course such a collision might break the molecules into smaller pieces; in that case, we would say that energy of motion had been converted into a form of internal energy). Collisions of these and many other sorts are going on all the time, at a numerically immense rate. In each cubic inch of air near the surface of the Earth there are many billions of trillions of molecules, and each of them undergoes millions of collisions per second. With so many collisions keeping track of individual results becomes meaningless, and only a calculation of the average energy of the particles is of much value.
      Such a calculation was carried out, based on the idea that the collisions should be completely 'random' or unpredictable in their individual details, almost a century and a half ago by Maxwell and Boltzmann. As a result of their calculations, they found that the molecules have a broad range of speeds, ranging from nearly zero (as in the example of a collision involving particles going in opposite directions), through the average speed (or speed of sound) and up to, rather surprisingly, speeds several times faster than the average speed. A comparison of the speeds or energies of the various particles to the number of particles with a given speed or energy is referred to as the Maxwell-Boltzmann Distribution. Here is a graph showing the results of such a calculation:

The number of particles (plotted upwards) which have a particular velocity (plotted to the right) peaks at a value not far from the average speed (the speed of sound), but there are some particles that are moving much slower or much faster than that at any given time.

     In the diagram above the speeds of particles are plotted from left to right, with particles of zero speed on the far left and fast-moving particles on the far right. The number of particles with a given speed is plotted vertically, with lower numbers of particles at the bottom and greater numbers at the top. The highest part of the curve, near the left, represents the 'average' speed, which is close to the speed of sound. The long, gradually declining part of the curve on the right is called the 'high-velocity tail', and represents particles that are moving much faster than the average speed.

Motions of Particles in a Gas: The Shuffling of Speeds Through Collisions
      In some ways it is quite surprising that the particle speeds in a gas have such a wide range. After all, if a particle is moving much faster or slower than the average speed of the gas particles, its next several collisions, most of which will be with particles which are moving close to the average speed, should cause the particle with an unusual speed to end up with a more average speed. After a while you might expect that all of the particles would in this way end up with nearly average motions; and yet according to very direct and (for those who are used to using calculus) relatively simple calculations, it is absolutely certain that the range of speeds shown in the Maxwell-Boltzmann distribution must exist at all times, and that this range of speeds is what is actually 'normal'. A gas which did not have such a range of speeds would actually be quite abnormal.
      But if particles which are now moving slowly will soon be moving faster, and particles which are now moving quickly will soon be moving slower, how can the Maxwell-Boltzmann Distribution of speeds be maintained? Simply put, by having particles which are now moving at a nearly-average speed end up, willy or nilly, with much slower or faster speeds. It is very easy to show, even without any mathematics, how some collisions could result in abnormally slow speeds. It is much harder to show without mathematics why some collisions will inevitably result in some particles moving much faster in the future than they are now moving; but it is absolutely certain that this is exactly what happens.
      As a result, we must not think of the Maxwell-Boltzmann distribution as representing the speeds that particles have both now and in the future. On the average, the distribution stays exactly the same. But there is a continual shuffling of speeds through the virtually infinite number of collisions which are occurring at all times. No matter what speed a particle now has, it will at some time in the future have various speeds, sometimes slow, sometimes fast, and sometimes average. For the particles which are now moving quickly there is a very high probability that they will soon be moving at much more average speeds. For the particles which are now moving at an average speed, there is a fairly substantial probability that they will soon be moving at almost-average speeds, but there is also a small probability that they will soon be moving much slower or much faster. There aren't very many particles which are moving very fast or very slow, so it only requires a small fraction of the currently average-motion particles to change to less average speeds in order to maintain the average distribution of speeds.

The Loss of Planetary Atmospheres
      At the top of a planet's atmosphere particles are running around in all directions at all of the various speeds corresponding to the kinetic temperature and the predictions of the Maxwell-Boltzmann distribution. Some of the particles will be headed upwards, some downwards, and some sideways. Some of them will be moving slowly, some at an average speed, and some very quickly. Whether a planet will hold onto an atmosphere depends upon the motions of those particles which happen to be moving upward at a much higher than average speed. If those particles are moving upward at less than the planet's escape velocity (the speed which an object must be traveling at in order to escape the planet's gravity and go off into space), the particles will follow curved paths which are ellipses with a focus at the center of the planet, and will go up for a while, then fall back into the atmosphere. (This discussion assumes that we are in the very outermost reaches of the atmosphere, where there is so little gas that the particles don't collide with other particles very often. If we were talking about a lower region the particles would be deflected from their paths and change their energies so frequently that any discussion of motions which resemble orbital motions would be pointless.)
      However, if the particles were moving upwards faster than the planet's escape velocity, they would follow hyperbolic paths which would take them out into space, never to return. Of course, only those particles which happened to be heading upwards at very high speeds would follow such paths, but as already discussed, there is a continual shuffling of particle motions and speeds, and as a result, in a short while particles which did not originally have such motions would end up with motions identical to those particles which had been lost, and then those particles would also be lost. Eventually, most of the outer atmosphere of the planet would disappear into space, and the lower parts of the atmosphere, which were previously held down by the (very slight) weight of the outer atmosphere, would expand and replace the missing parts of the atmosphere, and in a similar way would eventually be lost.

The Rate of Atmospheric Loss
      How fast the atmosphere of the planet is lost depends on how the average speed of exospheric particles compares to the escape velocity of the planet. Although the high-velocity tail of particle speeds can extend to many times the average speed, the number of particles in the tail becomes smaller and smaller as you go to higher and higher speeds. If the escape velocity is many times larger than the average speed, so few particles escape at any given time that it takes immense time periods for any gas to escape; whereas, if the escape velocity were a little lower, so that there were more particles which had enough speed to escape, the time required to lose a given part of the atmosphere would be considerably less.

      (due to lack of time when I was writing the above discussion, this is its current end; the following provides a summary of the eventual end of the discussion)

Summary of How Particle Velocities Compare to Escape Velocity
(quick notes from a class lecture, to supplement the above discussion)
     At the surface of the Earth the speed of sound (approx = average particle speed) = 1/5 mi/sec, but the speed depends upon the temperature, and at the top of the atmosphere (in the thermosphere) the temperatures are much higher than at the surface:
     At "room" temperature (70 F = 530 F above Absolute Zero), the speed of sound is 1/5 mi/sec. Temperature is proportional to the square of the average particle speed, so doubling that to 2/5 mi/sec would increase the temperature to 2100 F above Absolute Zero, or about 1650 F, which is close to the average temperature in the thermosphere, and less than its maximum temperature (near and during solar maximum).
     So in the exosphere (the outer part of the thermosphere), the average speed of "normal" air molecules is between 2/5 and 1/2 mi/sec.
     This speed has to be compared to escape velocity (8 mi/sec) to decide whether particles can escape. The escape velocity is 16 to 20 times greater than the average velocity of normal molecules (8 divided by 2/5 or 1/2).
     The question of whether an atmosphere can escape is usually expressed in terms of the Jeans theory (by James Jeans, about 1916?). There seems to be some disagreement in the exact results, which are complicated, but the following is generally agreed upon: The rate of loss of an atmosphere depends upon the ratio of escape velocity to average particle speed in the upper atmosphere. IF THE RATIO IS 5, THE ATMOSPHERE ESCAPES IN TIMES OF THE ORDER OF 100 MILLION YEARS. (General agreement about this result).
     As the ratio goes down (4, 3, etc) the atmosphere escapes much more rapidly (100 to 1000 times more rapidly for each unit change, depending upon the reference consulted), and as it goes up (6, 7, etc) the atmosphere escapes much more slowly (100 to 1000 times more slowly, as per previous statement). Hence, times to most of the atmosphere to escape are:
     About 100 million years if the ratio of escape velocity to average particle velocity is 5.
     Well under 1 million years if the ratio is 4 (since there are more particles in the high-velocity tail).
     Well under 10 thousand years if the ratio is 3 (still more high-velocity particles).
     But over 10 billion years if the ratio is 6 (fewer high-velocity particles).
     And well over 1 trillion years if the ratio is 7 (still fewer high-velocity particles).
     SO NORMAL GASES IN OUR ATMOSPHERE, with ratios of 16 to 20, are held onto forever (as we might expect, since we're breathing them after 4.5 billion years of Earth history).
     HOWEVER, lighter gases move faster (at the atomic level) than heavy ones, because temperature is proportional to particle mass, as well as the square of their speed. Oxygen atoms move 40% faster than oxygen molecules, helium atoms move 4 times faster, and hydrogen atoms move almost 6 times faster, than the "average" speed of normal particles. Therefore, the ratio of 16 to 20 times for normal air molecules is cut to 11 to 14 for oxygen atoms (still high enough to hold on, forever and ever), 4 to 5 for helium (lost in times of the order of millions to tens of millions of years), and about 3 for hydrogen (lost in times of the order of thousands or tens of thousands of years).
     IN OTHER WORDS, the Earth can hold onto either molecular or atomic nitrogen and oxygen, but slowly loses helium and more rapidly loses hydrogen, so that over long periods of time all the hydrogen and helium in the atmosphere should slowly leak into space.
     Actually, the losses can be even greater, as this ignores the possibility (the substantial possibility, according to some calculations) that ionized particles can be accelerated along the magnetic field lines and into space at an even greater rate. For helium, in particular, the loss rate must be of the order of a million years or less (given the low helium abundance in our atmosphere and the rate of helium production by radioactive decay inside the Earth), which is considerably faster than predicted by these numbers, so that the loss of ionized helium particles along magnetic field lines in the Arctic and Antarctic is thought to be the primary loss mechanism.

A Quick Summary of How This Will End When Finished
      If a planet is very large, like one of the Jovian planets, so that its escape velocity is far larger than the average speed of the gas particles even for the faster-moving atoms of hydrogen and helium, then it will be able to hold onto any gas, including those "light" gases.
      If a planet is fairly large, such as the Earth and Venus, so that its escape velocity is far larger than the average speed of heavier gas particles such as oxygen and nitrogen atoms and molecules, then it will be able to hold onto those gases, but the lighter atoms of hydrogen and helium, because of their greater speeds, will be too close to the escape velocity to be indefinitely retained, and those gases will gradually leak away from the planet.
      If a planet is fairly small, or a moon of a planet is fairly large, such as Mars and Mercury or our Moon or Titan, then whether it can hold onto even heavy gases will depend upon the planet's size and escape velocity, and its atmospheric temperature. If it is a larger planet a little further from the Sun like Mars, it may be able to hold onto gases, whereas if it is a smaller planet closer to the Sun like Mercury, it may not be able to do so. Similarly Pluto, although small, is far from the Sun and very cold, so it can hold onto gases (as long as it is not so cold that they would be frozen), while our Moon, which is considerably larger, because it is much hotter (being much closer to the Sun) cannot hold onto gases. Keep in mind that whether an object has an atmosphere doesn't depend on just whether it can hold onto gases, but also on whether it develops one. Titan has a thick atmosphere, while Ganymede and Callisto, which are equally capable of holding onto an atmosphere, don't have one (see the discussion of Titan for a possible theoretical explanation of this difference).
      If an object is very small (less than about 1000 miles, or 1500 kilometers diameter), so that its escape velocity is also very small, it will not be able to hold onto any gases.