MadSci Network: Astronomy

Re: Why doesn't the Moon have an atmosphere like the Earth?

Date: Sun Feb 24 18:14:41 2002
Posted By: Andy Goddard, Staff, Teaching and Learning Resources, Strathclyde University
Area of science: Astronomy
ID: 1014299232.As

Hi James!

The answer has to do with gravity and heat. It's complicated, but 
essentially the heat is a result of solar illumination, while the gravity 
depends on the mass of the body. For the Earth and Moon the solar 
illumination is virtually identical.

The escape velocity for a planet or satellite is equal to the square root 
of 2 times the mass and gravitational constant, divided by the radius. For 
Earth, in SI units, this equals:

Vesc = SQRT (2*Mass*G/r)
Vesc = SQRT (2*5.976e24*6.672e-11/6378000)
Vesc = 11182 ms-1

Any particle achieving this velocity or more will be lost from the Earth 
forever, be it a space probe or molecule of atmospheric gas. For the Moon 
the equivalent velocity (data from is:

Vesc = SQRT (2*7.35e22*6.672e-11/1738000)
Vesc = 2376 ms-1

Now, the average speed of a molecule in the Earth's atmosphere reflects 
the temperature of the gas. The kinetic energy for a molecule is:

KE = 1/2 m*v^2

The average amount of thermal energy for a molecule in a gas is:

KE = 3/2 kb*T

Where kb is Boltzmann's constant, and T is the temperature in Kelvin. 
Since these equations are equal...

1/2 m*v^2 = 3/2 kb*T

We can rearrange for v:

v = SQRT(3*kb*T/m)

For oxygen, (m=5.3e-26 kg) at a solar induced temperature of 295 K, where 
kb = 1.38e-23:

v = SQRT(3*1.38e-23*295/5.3e-26) = 480 ms-1

Similarly hydrogen (m=6.625e-27 kg) equals v = 1365 ms-1

Now, a real gas contains a bell curve of velocities of its component 
molecules, all averaging to the above figures per molecule, but with some 
molecules slower, and others higher. Both the Earth and Moon have 
outgassed to produce atmospheres since their creation, and over time the 
fastest molecules (those that exceed v escape) will leach out of the 
atmosphere - so the Earth will lose hydrogen molecules faster than it does 
oxygen molecules, and with lower escape velocities at the Moon, the Moon 
will lose all gaseous elements faster than the Earth.

A further complication occurs as solar UV can break up molecules high in 
the atmosphere: for example, water vapour (H2O) can be dissociated into 
hydrogen molecules and monoatomic oxygen. The hydrogen, being lighter, can 
escape more easily, contributing to the very low presence of hydrogen in 
the Earth's atmosphere. At the Moon, even oxygen molecules are lost from 
dissociated H2O relatively rapidly.

I hope this helps!

Andy Goddard

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