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There is not enough gravity at the center to start nuclear fusion, but it seems that there would be plenty enough to collapse the planet.

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I am not a physicist and I am just guessing, but couldn't rotation of the star be the reason? –  Sebastian Langer Jul 17 '13 at 22:09
    
I'm inclined to agree... I do know that gas giants tend to spin very quickly. –  aserwin Jul 17 '13 at 22:13
    
Pulsar's answer makes more sense than what I said. If it was spinning without heat flux, it would still collapse, I guess? –  Sebastian Langer Jul 17 '13 at 22:31
    
    
Maybe it is falling on itself, it's just not free falling! –  Ali Jul 18 '13 at 8:15
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2 Answers

Jupiter's gravity is balanced by the thermal pressure of its atmosphere: Jupiter is in hydrostatic equilibrium (or quasi-equilibrium: it slowly loses heat in the form of radiation).

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Pulsar's answer is indeed correct, but let me expand a bit more.

What happens when a gas giant shrinks?

A uniform mass will have a self gravitational potential of $-\frac{3GM^2}{5R}$. If we decrease its radius, its potential will decrease as well and the difference will be turned into thermal energy. Although gas giants and stars are not uniform mass balls, their gravitational binding energy is still proportional to $\frac{GM^2}{R}$, Thus if the radius decreases it will release energy, which will raise the temperature in return.

What happens when the temperature increases?

Assuming the gas in those planets obey the ideal gas law $$PV=nRT$$ (where $R$ is not the radius but the molar gas constant $R=8.314\,\text{J K}^{−1}\text{mol}^{-1}$), it's obvious that when $T$ increases and $V$ decreases (due to the shrink in the previous section) $P$ must increase. Note that most real gases behave qualitatively like an ideal gas, so this is not a crazy assumption.

So what is the big picture?

The planet shrinks a little bit, the potential difference turns into thermal energy and its temperature rises. The rise in temperature will cause the pressure to rise and prevent the planet from shrinking further (holding the planet in hydrostatic equilibrium). However, the planet also loses energy due to EM radiation as well, so it will continuously shrink and radiate. The process is called Kelvin–Helmholtz mechanism.

For instance, Jupiter is shrinking the tiny bit of $2\,\text{cm}$ each year. Although you might think this is really nothing, the amount of heat produced is similar to the total solar radiation it receives.

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Could you give a reference for the 2cm a year shrinking? Do you know whether this rate is dominated by heat loss, or by the shrinking arising from space debris falling into Jupiter? –  WetSavannaAnimal aka Rod Vance Aug 3 '13 at 2:29
    
@WetSavannaAnimalakaRodVance That is a well known number. I think the Wikipedia page of Jupiter mentions it as well. I think it is dominated(even calculated) by heat loss. –  Ali Aug 3 '13 at 5:35
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