what is the minimum size required by an object so as to retain an atmosphere?

Question

like say the moon could it sustain an atmosphere?

but given the amount of gasses it would take to make up a substantial atmosphere i dont see why it couldnt have one. plus neither of you answered my question.

Answer 1

That depends on the temperature. And that depends on distance from the sun. A body the size of the moon gets too hot to retain an atmosphere. A body of the same size out past Jupiter or Saturn should have little problem in retaining some gasses.

Answer 2

Richard R gave a good answer.

Air molecules are continually "leaking away" from the atmospheres of all planets (including the earth). If a molecule is near the top of the atmosphere, all that is required for it to escape, is that it be kicked out (by random collisions) at a speed that exceeds the planet's escape velocity. This is more likely to happen if the planet is hot (because then the gas molecules move faster), or if the planet is small (because then its escape velocity is small).

So if a planet is hot, its atmosphere may escape to space. But on the other hand, if it's too cold, its atmosphere may just freeze out and drop to the ground.

On top of that, since all atmosphere's are somewhat "leaky," it also depends on how rapidly the atmosphere can be replenished by gases that come out from the surface. You would expect Venus to have a thinner atmosphere than the earth (because it's hotter and its escape velocity is about the same as earth's) but in fact its atmosphere is much thicker than ours; this indicates that there is (or has recently been) much outgassing from Venus' surface to replenish the molecules that escape into space. Likewise, Mars' atmosphere is known to have once been much denser than it is today; its thick atmosphere was probably lost due to geological changes that stopped replenishing it.

So, in general, it depends not only on the planet's gravity, but also on its temperature and its geological activity. In the case of the moon, there's virtually no geological activity, and it's too warm and too small. If you dumped a bunch of gases onto the moon, the sun would warm them up to the point where the molecules would exceed the moon's escape velocity, and the new atmosphere would leak away.

Answer 3

Molecules evaporate from the top of the atmosphere. The rate of evaporation into space is either significant or insignificant, depending on the mass of the particular kind of molecule, the strength of the planet's gravity, and the exosphere temperature.

I'll show you how to calculate whether a particular gas is "stable" in the atmosphere (evaporation half-life of 10 billion years) or unstable.

T = 393.6 K { (L/Lsun) / [ a(1-e) / (1 au) ]^2 }^(1/4)

where...
T = the planet's exosphere temperature
L = the luminosity of the star in watts
Lsun = 3.826E+26 watts
a = the semimajor axis of the planet's orbit
e = the eccentricity of the planet's orbit

For example, let's do Mars.

L = Lsun
a = 1.524 au
e = 0.0933

T = 393.6 K { 1 / [ 1.524 (1 - 0.0933) ]^2 }^(1/4)
T = 334.8 K

Now we find the "average" (RMS) speed of a species of molecule, as a function of its molecular weight.

Vrms = (157.5 m/s) sqrt { (T / 1K) / m }

where...

Vrms = the average speed
m = the molecular weight

Let's do five gases for Mars: methane (m=16), water (m=18), diatomic nitrogen (m=28), diatomic oxygen (m=32) and carbon dioxide (m=44).

Vrms (CH4) = (157.5 m/s) sqrt { 334.8 / 16 }
Vrms (CH4) = 720.5 m/s

Vrms (H2O) = (157.5 m/s) sqrt { 334.8 / 18 }
Vrms (H2O) = 679.3 m/s

Vrms (N2) = (157.5 m/s) sqrt { 334.8 / 28 }
Vrms (N2) = 544.6 m/s

Vrms (O2) = (157.5 m/s) sqrt { 334.8 / 32 }
Vrms (O2) = 509.4 m/s

Vrms (CO2) = (157.5 m/s) sqrt { 334.8 / 44 }
Vrms (CO2) = 434.5 m/s

Now, we find the escape speed from Mars.

Vesc = sqrt (2 G M / R)

where...
Vesc = the escape speed
G = 6.673E-11 m^3 kg^-1 sec^-2
M = the mass of the planet
R = the radius of the planet (to the top of the atmosphere)

M = 6.419E+23 kg
R = 3.39E+6 meters

Vesc = sqrt [ (2) (6.673E-11) (6.419E+23) / (3.39E+6) ]
Vesc = 5027.0 m/s

Now, we find the ratios of Vesc with respect to the average speeds for diatomic oxygen and for carbon dioxide.

Vesc / Vrms (CH4) = 5027 / 720.5 = 6.977
Vesc / Vrms (H2O) = 5027 / 679.3 = 7.400
Vesc / Vrms (N2) = 5027 / 544.6 = 9.231
Vesc / Vrms (O2) = 5027 / 509.4 = 9.868
Vesc / Vrms (CO2) = 5027 / 434.5 = 11.57

In order for a gas to be stable in a planet's atmosphere, at least from thermal agitation, the ratio Vesc/Vrms should be greater than 10. Or somewhere about there.

Of course, a gas can also disappear from an atmosphere because of photodissociation by the sun's radiation. It can break into smaller bits, and those smaller bits might escape or combine with something else in the air. Oxygen can deplete from the atmosphere by combining with minerals on the ground.

You can repeat this calculation for the moon, if you want to.

For the moon...
L = Lsun
a = 1.000 au
e = 0.01672
M = 7.348E+22 kg
R = 1.738E+6 meters

Answer 4

There isn't a real limit to this because there wouldn't come a point where there were no particles of gas which were not attracted by the object. The Moon has a very thin atmosphere, for example. Gravity does not have a limit to its range, regardless of its strength.

Answer 5

The Moon cannot sustain an atmosphere.
Mars barely.