3 ms·

This is one of those rare cases where one's immediate intuition isn't so far off. Globular clusters are very dense stellar environments. For context, the radi

This is one of those rare cases where one's immediate intuition isn't so far off. Globular clusters are very dense stellar environments. For context, the radius of a globular cluster is roughly the distance to the nearest star to our Sun (a few light-years). In this space there are packed roughly a million stars. So dynamical interactions are frequent. A consequence of this is that stars exchange energy with each other frequently, and when you have frequent energy exchanges, you end up with a system that is in energy equipartition --- i.e., every star has about the same amount of energy as every other. Since more massive stars have the same energy as lighter stars, they must move slower, and consequently sink to the center of the cluster. Ultimately energy equipartition is the same reason that there is gravitational settling in a gas or a liquid on Earth.

Tloewald 10y ago
Wonderful explanation. Thank you!
effie 10y ago
> when you have frequent energy exchanges, you end up with a system that is in energy equipartition Could you support this claim with some explanation? This can be proven for the Maxwell-Boltzmann distribution of velocities, which is valid for gas in thermodynamic equilibrium, but probably not so valid for stars as these interact via long range forces, not through collisions like molecules do. EDIT Also, the stars that move too fast will escape the system so the equipartition cannot be sustained for stars whose mass is lower than some value corresponding to average quadratic speed being equal to escape speed.
- antognini 10y ago
  I'm not sure if it has been proven mathematically. The statement that the long-term evolution of the system will tend towards equipartition is equivalent to the system being ergodic. Ergodicity means that statistically any particular state is as likely as any other. These systems are observed (both in reality and in simulation) to be consistent with being ergodic. But I'm not sure it's been proven. I'm also not sure that it has been proven that gasses are ergodic either, even though experiments are consistent with them being ergodic.
  - effie 10y ago
    > These systems are observed (both in reality and in simulation) to be consistent with being ergodic. How could one possibly conclude that system is ergodic from observation? I thought ergodicity is a property of infinite time averages. Globular clusters do not exist indefinitely in the same macrostate, they lose stars as they evolve.
    - antognini 10y ago
      This is not my specialty so I may be getting some details wrong here. I don't mean so much that globular clusters are ergodic as few-body interactions are ergodic. It has at least been proven that three-body interactions are chaotic, which implies that all microstates are accessible to the system. This is a necessary requirement for ergodicity, but I don't believe that it implies that the interactions are necessarily ergodic. All I really know is that in practice globular clusters are observed to obey equipartition of energy, as are the average outcomes of large numbers of few-body interactions.
    - hansen 10y ago
      Ergodicity in statistical mechanics roughly means that you can replace an average over time by an average over states. In practice this means that within the time it takes to perform your measurement the system should move through a lot of “typical” states. So for everything that matters: If statistical mechanics works you can conclude that the system is ergodic. And if you observe long-time correlations it’s not.
- Florin_Andrei 10y ago
  > probably not so valid for stars as these interact via long range forces, not through collisions like molecules do I'm guessing it should be close enough due to the inverse square root law. Just the shape of hyperbolic trajectories should provide a clue. > the stars that move too fast will escape the system so the equipartition cannot be sustained Again, I think this one of those "close enough" things. Reality tends to dislike either-black-or-white approaches.
  - effie 10y ago
    The inverse square root law is very different from intermolecular interaction. I do not see why the equipartition should be close enough to actual behaviour if the interactions are so much different. I am not sure how the shape of hyperbolic trajectories is of relevance for equipartition, since most of the stars in the cluster do not have such trajectories else it would disintegrate before it could achieve any sort of dynamical equilibrium.
- effie 10y ago
  Equipartition in globular clusters is controversial. Trenti M., Marel R.: No Energy Equipartition in Globular Clusters https://www.astro.umd.edu/~richard/ASTRO620/glob_clust_energy_equip.pdf https://www.astro.umd.edu/~richard/ASTRO620/glob_clust_energ...

Tloewald 10y ago

Wonderful explanation. Thank you!

effie 10y ago

> when you have frequent energy exchanges, you end up with a system that is in energy equipartition Could you support this claim with some explanation? This can be proven for the Maxwell-Boltzmann distribution of velocities, which is valid for gas in thermodynamic equilibrium, but probably not so valid for stars as these interact via long range forces, not through collisions like molecules do. EDIT Also, the stars that move too fast will escape the system so the equipartition cannot be sustained for stars whose mass is lower than some value corresponding to average quadratic speed being equal to escape speed.

antognini 10y ago
I'm not sure if it has been proven mathematically. The statement that the long-term evolution of the system will tend towards equipartition is equivalent to the system being ergodic. Ergodicity means that statistically any particular state is as likely as any other. These systems are observed (both in reality and in simulation) to be consistent with being ergodic. But I'm not sure it's been proven. I'm also not sure that it has been proven that gasses are ergodic either, even though experiments are consistent with them being ergodic.
- effie 10y ago
  > These systems are observed (both in reality and in simulation) to be consistent with being ergodic. How could one possibly conclude that system is ergodic from observation? I thought ergodicity is a property of infinite time averages. Globular clusters do not exist indefinitely in the same macrostate, they lose stars as they evolve.
  - antognini 10y ago
    This is not my specialty so I may be getting some details wrong here. I don't mean so much that globular clusters are ergodic as few-body interactions are ergodic. It has at least been proven that three-body interactions are chaotic, which implies that all microstates are accessible to the system. This is a necessary requirement for ergodicity, but I don't believe that it implies that the interactions are necessarily ergodic. All I really know is that in practice globular clusters are observed to obey equipartition of energy, as are the average outcomes of large numbers of few-body interactions.
  - hansen 10y ago
    Ergodicity in statistical mechanics roughly means that you can replace an average over time by an average over states. In practice this means that within the time it takes to perform your measurement the system should move through a lot of “typical” states. So for everything that matters: If statistical mechanics works you can conclude that the system is ergodic. And if you observe long-time correlations it’s not.
Florin_Andrei 10y ago
> probably not so valid for stars as these interact via long range forces, not through collisions like molecules do I'm guessing it should be close enough due to the inverse square root law. Just the shape of hyperbolic trajectories should provide a clue. > the stars that move too fast will escape the system so the equipartition cannot be sustained Again, I think this one of those "close enough" things. Reality tends to dislike either-black-or-white approaches.
- effie 10y ago
  The inverse square root law is very different from intermolecular interaction. I do not see why the equipartition should be close enough to actual behaviour if the interactions are so much different. I am not sure how the shape of hyperbolic trajectories is of relevance for equipartition, since most of the stars in the cluster do not have such trajectories else it would disintegrate before it could achieve any sort of dynamical equilibrium.
effie 10y ago
Equipartition in globular clusters is controversial. Trenti M., Marel R.: No Energy Equipartition in Globular Clusters https://www.astro.umd.edu/~richard/ASTRO620/glob_clust_energy_equip.pdf https://www.astro.umd.edu/~richard/ASTRO620/glob_clust_energ...