The Gravo-Thermal Catastrophe

V. A. Antonov did somes remarkable simulations showing that even in Newtonian mechanics, gravitating systems can violate the usual rules of thermodynamics. Instead of reaching equilibrium they get hotter and hotter!

Suppose you put a lot of stars in a large sphere, and suppose (unrealistically) that they bounce elastically off the walls of this sphere. In fact suppose they’re point masses, so they never collide, and interact only gravitationally. Also suppose they’re ‘gravitationally bound’. This means their total energy, kinetic and potential, is negative. That means they couldn’t all shoot off to infinity even if the sphere wasn’t there holding them in.

What happens?

If the sphere is small enough, the stars will seem to come into equilibrium, like a gas of roughly constant density. Antonov showed numerically that this happens if the radius R of the sphere obeys

where M is the total mass of the stars, E is their total energy, and G is Newton’s gravitational constant.

But the exciting thing is what happens if the sphere is larger than this!

In this case some stars will collect near the center and form a dense cluster. Eventually this cluster will start to collapse, and get very ‘hot’. That is: stars in the cluster start to move very fast. Some shoot out of the cluster at high speeds… but other stars fall into the cluster, replenishing it.

As a result, all the stars in the sphere keep moving faster and faster! The pressure on the sphere’s walls keeps increasing! This is the 𝗴𝗿𝗮𝘃𝗼-𝘁𝗵𝗲𝗿𝗺𝗮𝗹 𝗰𝗮𝘁𝗮𝘀𝘁𝗿𝗼𝗽𝗵𝗲.

I hope you see why this doesn’t violate conservation of energy: the increasing negative potential energy of the cluster balances the increasing kinetic energy.

To prevent the sphere from exploding, you may decide to make it larger. But this won’t help for long. The central cluster will keep getting hotter, and heating up all the stars around it.

What do you do? There’s only one way to stop this runaway temperature increase!

The only solution is to shrink the sphere, fighting against the pressure exerted by the gas of stars within. Once it’s small enough—about the size of the dense cluster of stars near the middle—you’ll be back to equilibrium (or so it seems). But the temperature and pressure now is much higher than when you started!

Furthermore, even when the gas of stars seems to be in equilibrium, it’s not. It’s just a metastable state. Eventually by chance a small dense cluster will form, and the gravo-thermal catastrophe will occur!

Here’s a great article about the gravo-thermal catastrophe, featuring one of the best uses of narrative I’ve seen in a physics paper:

• D. Lynden-Bell and Roger Wood, The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems, Mon. Not. Roy. Astr. Soc. 138 (1968), 495–525.

The best part is Section 2, where they vividly tell the tale of how you struggle to deal with a sphere full of stars as its temperature and pressure starts rising uncontrollably!

Note that when I say ‘star’ above, it’s just a dramatic way of talking about a massive point particle. But since stars rather rarely collide, the study of point particles interacting via Newtonian gravity helps us understand more realistic stellar clusters, like globular clusters. In globular clusters, the density often gets so high in the center that stars collide and form a black hole in the middle!

Here’s a globular cluster where this has happened now, called Messier 15:

What I’m saying is that from here on out, the situation will only get messier.