Re: How big would a planet need to be?

One of the big advantages of water as a substance in the environment is 
that it has a large liquid range. Its freezing point is just over 273 K, 
and at that temperature the saturation vapour pressure is just over 4 torr, 
or 0.006 atmosphere. At 1 atmosphere, liquid water boils at just over 373 
K, and at 20 atmospheres the boiling point is around 486 K.

If we look at methane instead, the freezing point is just 90.5 K, and the 
vapour pressure at that temperature is about 90 torr, or 0.12 atmospheres.
At 1 atmosphere methane boils at 111.5 K, and at 20 atmospheres the boiling 
point is around 164.5 K.

We will round out the table with possible inert background gases:

gas     condensation temperatures at various pressures
             100 torr          1 atm          20 atm

nitrogen     63.5 K            77.4 K         115.6 K
argon        72.6 K            87.6 K         131.9 K
neon         22.2 K            27.2 K          42.4 K
hydrogen     15.3 K            20.7 K         supercritical
helium        2.8 K             4.6 K         supercritical

There really are no other possibilities, and even among these, neon does 
not have a high planetary abundance on either inner or outer planets, for 
good geochemical reasons.

That data comes from the CRC Handbook of Chemistry & Physics. We will 
return to it a bit later.

Now let us look in more detail at your specific questions.

(1) Whether or not a planet can capture and hold a gas in its atmosphere 
depends on 
    a.) the planetary size (which governs the strength of its gravitation)
    b.) the planetary temperature, and in particular the detailed 
temperature profile of its atmosphere (which governs whether or not 
molecules are likely to be moving fast enough to escape its gravitational 
field)
    c.) the intensity of solar radiation in the outer atmosphere, and what 
particular shielding and reaction mechanisms prevail there (which governs 
whether a molecule like methane, which is too heavy to escape, will be 
changed by photochemical reactions into a molecule like hydrogen, which is 
not!)

In this immediate part of the solar system, the prevailing temperature and 
radiation regimes are such that an Earth-size planet can not hold hydrogen, 
narrowly fails to hold helium (It takes an average of 1 million years for a 
helium atom to escape the gravitational field of the Earth from the inner 
atmosphere), and has no difficulty in holding anything else indefinitely, 
as far as simple gravitational loss is concerned. A planet would need to be 
smaller than the moon to run any real risk of losing heavier gases to 
simple gravitational effects. (everything else is heavier than methane, 
water, and ammonia, which are 4 times heavier than helium, which in turn is 
twice as heavy as hydrogen). Of course it is harder for a planet to hold 
its atmosphere when it gets really hot, and easier for really cold planets.

The real problem with a planet not holding methane is that ultraviolet 
radiation can break up a methane molecule into a methyl radical and a 
hydrogen atom. If pressures are at all high, these species are short-lived, 
but reactions can follow which produce ethane and molecular hydrogen, and 
this molecular hydrogen is subject to gravitational loss from an Earth-
sized planet.

(2) On the Earth, surface temperatures average around 288 K, and range 
between about 230 and 320 K. Solid water exists at the lower part of the 
range, and liquid water at the upper part, provided that the atmospheric 
pressure is greater than about 0.1 atmosphere (320 K), 0.01 atmosphere (280 
K). Water vapour pressures range from less than 1 torr in Arctic regions up 
to about 30 or 40 torr in the humid tropics.

For methane to exist as a liquid, the temperature would have to be at least 
91 K. If the atmospheric pressure were only 1 atmosphere, it could only 
range up to about 108 K before all of the liquid methane would be lost in 
evaporation. An atmospheric pressure of 20 atmospheres would get us up to 
150 K or so.

(3) Could nitrogen co-exist as a gas? Only if the temperature range was 
quite narrow. For a 1 atmosphere nitrogen atmosphere, the temperature range 
would be restricted to 78 - 90 K where solid methane could exist, and 91-
108 K where liquid methane could co-exist -- a total temperature range of 
only 30 degrees.
Could argon co-exist? No, the range would be quite impractically narrow for 
solid methane.
Could any other gas? Yes, neon, but it is not easy to see how a planet 
could evolve in such a way that neon would not be rare. Hydrogen and helium 
are also possibilities for large planets and low atmospheric pressures.