https://johncarlosbaez.wordpress.com/2024/09/20/the-gravo-thermal-catastrophe/ Azimuth * Home * About The Gravo-Thermal Catastrophe In 1962, V. A. Antonov did some remarkable simulations showing that in Newtonian mechanics, gravitating systems can violate the usual rules of thermodynamics. Instead of reaching equilibrium they can 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 R < 0.335 GM^2/(-E) 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 shrinking cluster balances the increasing kinetic energy of the stars whizzing around faster and faster. It's the arbitrarily large negative potential energy of this system that makes it break the usual rules of thermodynamics. It can keep converting potential energy to kinetic energy, endlessly. A much simpler example would be a bunch of balls rolling down an endless ramp. To prevent the sphere from exploding due to the ever-rising pressure inside, you may decide to make it larger. This will help--but not for long. The pressure will drop at first. But the central cluster will keep getting hotter, and heating up all the stars around it. Eventually the pressure will rise again. 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 tremendous pressure exerted by the gas of stars within. Once the sphere is 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, called Messier 15: [m15_globular_cluster_from_the_mount_lemmon_skycenter_schulman_t] What I'm saying is that from here on out, it will only get messier. [gravo-thermal_catastrophe-1] Related This entry was posted on Friday, September 20th, 2024 at 7:46 pm and is filed under astronomy, information and entropy, physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. Post navigation << Previous Post 6 Responses to The Gravo-Thermal Catastrophe 1. [c52] Yuan says: 21 September, 2024 at 4:02 am Is this phenomenon related to the works on the stability of matter by Dyson, Lieb, Thirring, and others? Basically, the model is thermodynamically unstable because there is no "repulsive core"? Reply + [7a9] John Baez says: 21 September, 2024 at 4:29 am It's interesting to compare this to the work of Dyson, Lieb, Thirring and others on the stability of matter. They proved that for N charged quantum particles interacting electostatically, the lowest possible energy, which is negative, is bounded below by a function that's linear in N. For classical particles interacting gravitationally things are much worse. For finite N there is no lower bound on the energy! The main differences in the assumptions are that: 1) like masses attract while like charges repel, 2) the gravitational system is being treated classically while the charged particles are being treated quantum-mechanically, 3) the gravitating masses are assumed to obey Boltzmann statistics while the charged particles are assumed to obey Fermi-Dirac statistics. It would be good to understand all 2 × 2 × 2 variations on this stability question. I believe that the Fermi-Dirac statistics are necessary for the Dyson-Lieb-Thirring result: i.e. for N bosonic charged particles treated quantum-mechanically, while there would be a lower bound to the energy for any fixed N, this lower bound would be worse than linear in N. I don't know the story for gravitationally attracting quantum particles, but they must have been studied. Reply 2. [2cc] Gitel says: 21 September, 2024 at 6:44 pm Isn't it a perpetuum mobile of a kind? Reply + [7a9] John Baez says: 21 September, 2024 at 8:48 pm Yes! As mentioned, the usual laws of thermodynamics don't apply to systems where the potential energy can be arbitrarily large and negative. We can, if we want, conclude that such systems are 'unrealistic'. A realistic cluster of stars, somehow contained in a large sphere, would behave in a similar way at first, but not forever: eventually they would collide and form black holes, and cease to speed up. Reply 3. [fb2] Rick Roesler says: 22 September, 2024 at 2:45 am A messier Messier. Still laughing about this. Reply 4. [584] Sir Light says: 22 September, 2024 at 11:21 am But point masses do not exist, which does put an upper bound on the total energy the system could possess before the masses break down and act under other forces than gravity, doesn't it? Reply You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word 'latex' comes right after the first dollar sign, with a space after it. Cancel reply [ ] [ ] [ ] [ ] [ ] [ ] [ ] D[ ] This site uses Akismet to reduce spam. 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