The Fate of Gaseous Planets: Can They Stay Stable Forever?

[fbs7]

Hello all,

Sorry if the question sounds idiotic, but this morning I was watching a program on Jupiter and then I couldn't wrap my head around this. Say a slowly changing gas planet that is cold enough to have very little convection, and that by assumption does not lose its mass to space. Then draw some radius around a fixed mass M1 around it center (say 1 trillion tons); that will have a volume V which is very slowly changing. Then I know that pV = nRT. Now consider the rest of the gas outside that volume; that that's M2 = 1 quatrillion tons.

Now it gets very complicated for me, and here's the thinking. The planet will always lose heat through radiation; so let 1 trillion years pass. Whatever T it started with, in the long term T must end up being very small (for the reason that a higher T will lead to more heat loss due to radiation).

As R is constant and under my definition n is constant too, then pV is going to be very small. But there is a huge weight of the fixed mass M2 pushing down through gravity on my volume. So p cannot be very small. Therefore, V must be very small, and given that the radius of the volume is very small and the mass M2 is fixed, then p must be large.

As T is very small and p is large, then the core of a large enough gas planet must at some point freeze or liquefy. Therefore, above some size it should be impossible for a gas planet to remain gaseous forever. But that sounds preposterous, because it means that in the long term there would be no more large, completely gaseous planets, for thermodynamics alone...

I appreciate any insight where may train of thought has gone off tracks on that.

 

[DaveC426913]

Are you factoring Jupiter's source of heat? It generates radiation as it very slowly shrinks due to gravity.
If you are not factoring this in, it seems you need to assume it is possible to have a Jupiter-like planet that is not shrinking and producing its own heat.

 

[fbs7]

This was my train of thought - if the planet shrinks, then it heats up. But, whatever heat it generates, it will eventually be lost as radiation in the trillion years timeframe.

So, after a long enough time, T must necessarily end up being very low, as whatever process that generates heat will stop someday, while the heat keeps escaping forever as radiation. I just checked at what temperature hydrogen solidifies, and that's a few atm at something like 20 Kelvin. So I suppose that any ball of hydrogen that can generate a pressure of a few atm at its center will eventually freeze out someday as it loses heat... I guess...

 

[DaveC426913]

It is theorized that Jupiter has a solid core, though it is more due to pressure than temperature.

 

[snorkack]

As T is very small and p is large, then the core of a large enough gas planet must at some point freeze or liquefy. Therefore, above some size it should be impossible for a gas planet to remain gaseous forever. But that sounds preposterous, because it means that in the long term there would be no more large, completely gaseous planets, for thermodynamics alone...

It's fundamentally impossible for a gaseous planet of any size to be stable, period. Yes.
Consider a self-gravitating ball of any substance, and for simplification assume that the density is constant (Actually impossible for any fluid, but that simplification will be relaxed later on).
And now shrink its radius R some number, say 2 times.
What happens to its central pressure?
The surface gravity will increase 4 times.
The density will increase 8 times.
Therefore the weight of a column of given length will increase 32 times.
The length of the column from surface to centre has shrunk 2 times.
Thus the pressure at the centre has increased 16 times.
This above reasoning holds no matter whether the ball consists of gas, liquid or solid.
If the density isn't constant from surface to centre (as stated, it is impossible for fluids), but the radial distribution of density remains unchanged as the ball shrinks, then the scaling of central pressure still holds.
Therefore, pV=nRT is impossible. The central density has increased only 8 times, so isothermal gas should only increase its pressure 8 times, not 16 - and should collapse further.

 

[fbs7]

Oh, that's very impressive.

So heat generation is the only thing that keeps a gas planet from collapsing. Once the heat is gone, collapse is inevitable.

Most amazing thoughts. Thanks both for your insights.