QUESTION:
I'm puzzled by the gravity vs. atmosphere amounts of Mars compared to
Earth's. I recall the gravity is 40% of Earth's gravity, but the atmosphere
is only 10% as dense as Earth's.  My expectation was that the atmosphere
would also be at least 40% of earth's atmosphere because gravity affects the
amount of atmosphere present. Also carbon dioxide is a denser molecule than
nitrogen, so an inference might be that the atmosphere would be even more
dense than 40% of earth's. Does the temperature affect the relationship
between planet mass and atmosphere or are there other variables that I just
have not considered?

ANSWER from Jim Murphy on August 22, 1997:
The gravity vs. atmosphere correlation does not work quite as straight
forwardly as you might first think.  The best example of this might be the
difference between Earth and Venus.

Earth and Venus are approximately and have (I think) similar gravitational
pull at their surfaces.  However, the surface pressure on Venus is 90 times
the surface pressure here at the Earth's surface!  Thus, it is not strictly
gravity alone which affects the pressure.

As you point out, the differences in molecular weights of different gases
will have an effect upon the pressure (i.e., carbon dioxide molecules are
more massive than nitrogen molecules).  If we were to take a one meter
square column of atmosphere from Earth (with a surface pressure of 1000
millibars, this would amount to 10,000 kilograms of air; multiply 1000
millibars by 100 to convert to (mks) units of Pascals (units of kg per meter
per second squared), which when divided by gravity (meters per sesond
squared) leaves kilograms per square meters).  If I then took this column of
Earth atmosphere and placed it on Mars, Mars' smaller gravitational
attraction (40 percent of Earth's, as you pointed out) would result in a
pressure of only 400,000 Pascals, which is 400 millibars, or 40 percent of
the columns surface pressure on earth.  If the air molecules in our column
were all converted to carbon dioxide, each of which are 1.57 times the mass
of an 'air' molecule (44/28), than the surface pressure of that column on
Mars would increase to 628 millibars, and on Earth it would be 1570
millibars.

Temperature does not directly affect the relationship between planet mass
and atmosphere.  The warmer an atmosphere is, the 'deeper' it is (that is,
pressure decreases less rapidly with height in a warm atmosphere than in a
colder atmosphere).  The molecules at the top of this deeper (warmer)
atmosphere feel a slightly (very slight!) reduction in the gravitational
pull of the planet, and thus are just slightly more susceptible to processes
which might strip them from the planet.

What really determines the surface pressure of an atmosphere, in addition to
the planet's gravitational attraction, is how much gas is present.  The case
of venus vs. Earth really illustrates well the point that it is not the
planet's gravitational attraction alone which is important, but also how
much gas is there.  Earth could hold onto an atmosphere as massive as that
which Venus has, but for whatever reason (and there are several
hypotheses..), processes which have occurred over the past 4 billion years
or so have conspired to leave Venus in a state of a massive atmosphere,
Earth with a modest atmosphere, and Mars with a small mass of atmosphere.

So, a planet's (or moon's) gravitational attraction does play some role in
determining how massive an atmosphere that body might have, other processes
are more or equally important.

You have aterrific question.  I'd be interested in knowing what type of
explanations your student might come up with when discussing this issue.

Jim Murphy
Mars Pathfinder ASI/MET Science Team

ANSWER from Smart Filter on August 20, 1997:
Whether a planet can retain an atmosphere depends on several factors. Mass
and radius are important in determining escape velocity for the planet
(earth's is 11.2 km/sec; mars' is 5.0 km/sec.).  Temperature and the mass of
the molecule in question are the other two important factors, as those two
quantities are used in determining the average speed of the molecules in a
gas.  Molecular oxygen (O2) at 20 degrees Celsius has an average molecular
speed of 0.48 km/sec.  A rule of thumb is that a planet can retain a gas if
the escape speed is at least six times greater than the average speed of the
molecules in the gas. A lighter molecule than oxygen (like hydrogen) will
have a higher average molecular speed (the same calculation done for
hydrogen gives an average molecular speed of 1.9 km/sec).  A gas at a higher
temperature will also have a higher molecular speed.  So, you can see that
in addition to the size of the planet, we also have to consider distance
from the sun, and gases in the atmosphere. 

[My reference source for the above material is the text "Universe" (fourth
edition) by William Kaufmann published by W.H. Freeman and Co. copyright
1994, pp 134-135. You can look there for more information, including the
equations used in the calculations.]

Also, we have assumed thus far that nothing catastrophic happens in the
development of the atmosphere.  Astronomers are coming to believe that comet
and meteorite impacts can have a major effect on the atmospheres of
planets--both in delivering gases like water to the atmosphere, and in
stripping away much of the pre-existing atmosphere.