Why do some rocky planets and moons (Titan, Earth, Venus, Mars) have some sort of atmosphere and others none?

If a planet/moon's has an escape velocity greater than 10 times the mean velocity of the atmosphere's gas molecules, it will retain its atmosphere.The molecules in a gas have a range of velocities so the escape velocity of a planet must exceed the mean gas velocity by a factor of 10 to hold on to the gas, otherwise the faster velocity (higher temperature) gases will escape and slowly but surely the atmosphere will escape. The velocity of the gas molecules increases with temperature. Mercury's high surface temperature makes it impossible it to hold on to an atmosphere. According to Wikipedia, Mercury's escape velocity (Ve) is 4.25 kilometer per second, it surface temperature on the sunlight side can reach 700K (800°F). Titan which has an atmosphere has an escape velocity (Ve) of 2.64 kilometers per second. Titan's surface temperature 97 K (-180°C = --290°F)

The acceleration constant g equals

g = GM/r² Where G is the gravitational constant, M the mass in kilograms, and r the radius in meters

the escape velocity Ve = √(2gr)

The thermal velocity of a gas is proportional to √(T/m) where T is the temperature of the gas molecule in Kelvin, and m is the mass of the gas

It's a combination of things. First is the mass. An object must have enough mass to have enough gravity to attract the gases. Second is the presence of gases. This would mean the actual elements that make up the gases and the temperature to put them into a gaseous state. The gases could be trapped below the surface and be expelled by volcanic action, they could exist in a frozen state and eventually become airborne if the temperature rose high enough, or they could be brought to the planet via comets (where the source of the water on our planet is thought to have come from). Third would be proximity to the sun. Mercury has very little atmosphere, because it gets blown away by the solar winds before the atmosphere can really develop.

Good question. We really don't know. Titan is very similar to the moon Ganymede yet Titan has an atmosphere and Ganymede doesn't.

We can surmise that the existence of an atmosphere is the difference between when gasses are produced by outgassing or by bombardments by meteorites and comets and by how much gasses are lost due to photolytic action from the Sun. It's quite possible for moons and Planets to have atmosphere's temporarily as an influx of gasses perhaps from a period of volcanic activity or bombardments by comets and asteroids slowly subsides. To an extent all the moons have some atmosphere, our own Moon is said to barely have a measurable atmosphere from the outgassing of it's crust ( it's 0.3 nPa ).

P.S., Titan has 2.2% the mass of Earth while Mercury has 5.5% the mass of Earth. Titan is indeed half the mass of Mercury and has an atmosphere with twice the surface pressure of Earth's. It's not strictly based on the mass of the planet.

v = planet's escape speed, m/s

T = subsolar temperature of space near planet, Kelvins

M = planet's mass, kilograms

R = planet's radius, meters

L = sun's luminous power, watts

k = 1.3806488e-23 J K⁻¹

G = 6.67384e-11 m³ kg⁻¹ sec⁻²

σ = 5.670373e-8 W m⁻² K⁻⁴

m = the lightest atomic or molecular species that a planet can retain for 5 billion years

v = √(3kT/m) = √(2GM/R) / 6.7

3kT/m = GM/(22.445R)

m = 67.335kTR/(GM)

T = ∜[L/(4πσd²)]

m = 35.763367 k R ∜[L/(σd²)] / (GM)

@ = "appropriate MKS units"

m = (4.7945e-10 @) R ∜L / (M√d)

For sun and Earth,

R = 6378000 meters

L = 3.846e26 watts

M = 5.976e24 kilograms

d = 1.496e11 meters

m = 5.86e-27 kilograms

m = 3.53 amu

Helium would be almost gone from Earth's atmosphere after five billion years. All heavier atomic and molecular gases would be stable versus solar thermal evaporation. However chemical processes, such as oxidation, can also remove a gas from an atmosphere.

The number 6.7 in

v = √(3kT/m) = √(2GM/R) / 6.7

is a guess that thermal stability for five billion years requires that the escape speed from the planet's surface be greater than 6.7 times the RMS speed of a gas species of atomic weight m at the subsolar temperature T. Feel free to adjust, if you discover good empirical or theoretical reasons for doing so, and proceed to change the results accordingly.

It's simply a matter of the planet's mass. Below a certain mass, the planet's gravity isn't strong enough to retain an atmosphere. Ganymede is just about at the tipping point: slightly less massive than Titan, slightly more massive than Mercury.

Gravity, or the lack of it.