Expert: Philip A. Stahl - 9/10/2011

Question
Titan, one of Saturn's many moons, has an atmosphere, but our own Moon does not. What is the explanation?

Answer
Hello,

The bottom line is there's no easy explanation. ON the face of it, given Titan is about the same size as the Moon, and even has less density (1880 kg/m^3 vs. 3340 kg/m^3) and nearly the same mass (1.34 x 10^23 kg  vs. 7.35 x 10^22 kg) it seemingly a head scratcher. Especially also given the escape velocities are nearly the same.

In general, the condition for an object of mass M to retain an atmosphere whose molecules have mass m each, is:

6 [3kT/m]^½ < [2GM/R]^½

where T is the ambient temperature, k is Boltzmann's constant (k = 1.38 x 10^-23 J/K), G is the Newtonian gravitational constant (6.7 x 10^-11 Nm^2/kg^2) and R is the radius of the planet or Moon.

We already have the respective masses, and the radius of Titan is 2575 km, and of the Moon, 1727 km. The ambient temp. for Titan = 93.7 K and for the Moon (220 K, mean)

Now, molecular Nitrogen is the main constituent of the Titan atmosphere, so:

m = 14 x (1.7 x 10^-27 kg) = 2.3 x 10^-26 kg

Then:

6 [3kT/m]^½  = 6 [(3 x (1.38 x 10^-23 J/K)(93.7K) / (2.3 x 10^-26 kg)]^^½


6 [3kT/m]^½  = 2.46 x 10^3 m/s

And the escape velocity would be:

[2GM/R]^½ = [2 x (6.7 x 10^-11 Nm^2/kg^2) (1.34 x 10^23 kg)/ (2.57 x 10^6 m)


[2GM/R]^½ = 2.64 x 10^3 m/s

So the condition *holds* for TITAN.

Now, for the Moon:

6 [3kT/m]^½  = 6 [(3 x (1.38 x 10^-23 J/K)(220 K) / (2.3 x 10^-26 kg)]^^½

= 3.77 x 10^3 m/s

vs.

[2GM/R]^½ = [2 x (6.7 x 10^-11 Nm^2/kg^2) (7.35 x 10^22 kg)/ (1.72 x 10^6 m)


= 2.39 x 10^3 m/s

So, in this case, the condition for retention would not hold! I.e. the Moon would not be able to retain an atmosphere of molecular Nitrogen like Titan.

The conclusion then is that when we explore at more detail, using the physics for escape of an atmosphere, we find the Moon loses or would lose since its escape velocity is too low to retain it.

One would also find that just about any conceivable gas (e.g. H2, He, O2, etc.) would violate the condition for the Moon to retain. In each case the gas molecular speeds would exceed the Moon's velocity of escape.

What this shows is that temperature is the deciding factor. The relatively higher temperature for the Moon (220 K, mean vs. 93.7 K for Titan) yields a [2.3]^1/2 or 1.5 factor difference in the molecular velocity which pretty well ensures it will exceed the escape velocity thus violating the condition for retention.