Comment by antognini | Hacker News Reader

Comment by antognini | Hacker News Reader

To provide a little bit of context, it is known that globular clusters should produce a lot of black holes, but it has long been believed that they should retain very few, if any of these black holes. One reason for this is that black holes should receive a kick upon their formation. The gravitational potential of a globular cluster is weak enough that this would generally be enough to kick the black hole out of the cluster.

For those that remain in the cluster, the black holes would be much more massive than the rest of the stars in the cluster. Globular clusters undergo a phenomenon called mass segregation, where the most massive stars settle to the center of the cluster and lighter stars occupy the outer regions. So over time the black holes would sink to the center of the cluster where they would interact and form binaries with each other. These binary-binary interactions would, over time, kick single black holes out of the cluster. Since these interactions happen on timescales that are much shorter than the lifetime of the cluster, the received wisdom about twenty years ago was that globular clusters should have only a couple of black holes left, if any at all.

In 2007, Thomas Maccarone (who is an author on this paper) discovered a black hole in a globular cluster in another galaxy. It wasn't until 2012 that any were discovered in a globular cluster around the Milky Way. Now that computers have gotten fast enough that it has started to become possible to simulate the dynamics of globular clusters, theorists have found that globular clusters retain a much higher fraction of black holes than they originally thought. The reason for this seems to be that the distribution in black hole masses leads the black hole population to remain better mixed with ordinary stars than was originally thought.

> Globular clusters undergo a phenomenon called mass segregation, where the most massive stars settle to the center of the cluster and lighter stars occupy the outer regions.

Intuitively, this seems simple, since "heavier objects" should sink further down, no?

But thinking about this for a while, it doesn't seem so simple anymore. No matter what's the mass of the star, if it's already on a given orbit around the cluster, it will tend to stay on that orbit, unless there are close interactions with other stars of comparable mass.

So what would be an intuitive explanation for this phenomenon?

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.

Wonderful explanation. Thank you!

> 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.

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.

> 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.

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.

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.

> 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.

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.

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_energ...

A heavier star pulls the cluster towards it, as well. For an individual star this effect is small, but for many stars in a cluster this causes the heavier ones to begin to clump together.

Even if it leads you to the right conclusion, the intuition is a bit off: when something floats or sinks, it is floating or sinking in a medium. In the vacuum of space, there is no such tendency: all mass exerts an attractive force on other mass.

(however I have no idea how/why the mass segregation happens in this instance)

How does this "kicking" of black holes happen? Who is this galactic Henrik Lundqvist who keeps the black discs out of the cluster?

I'd guess that if a lower mass orbits a higher mass body in a very tight orbit it might get thrown out.

There are two ways that black holes could get kicked:

1. Natal kicks. The process of forming a black hole is fairly violent, and could involve a supernova in many cases. If the supernova is asymmetric, this will result in a kick on the black hole. Even if there is no supernova, an asymmetric infall process could produce an asymmetric burst of gravitational waves. Since gravitational waves carry momentum, this will also result in a kick on the new black hole. But it's generally thought that black hole natal kicks are weaker than neutron star natal kicks. The received wisdom seems to be that natal kicks will remove some black holes from a globular cluster, but probably not most.

2. Few-body interactions. A general principle of dynamical interactions (called Heggie's Law) is that if a single star interacts with a binary system, it is likely that the binary will become tighter if its orbital velocity is greater than the velocity of the incoming star, and wider if its orbital velocity is less than the velocity of the incoming star. The BH-BH binaries that would form in the cores of globular clusters would be relatively tight, so almost all interactions they have with other black holes will result in them getting tighter and giving more kinetic energy to the interloping black hole. (Or there may be an exchange and the interloping black hole will take the place of one of the black holes in the binary and kick out one of the original black holes.)

Back when I was in grad school I made a visualization of a binary-binary interaction, which you can watch here: https://www.youtube.com/watch?v=Y3nPOQXl6co

I've done n-body simulations like that in college. It's amazing to watch how, in certain cases, some random star, previously well-behaved, just shoots out of the cluster at high velocity after some unlikely chain of close encounters. The deserters extract some energy from the cluster when they leave.

Thanks for the explanation and the video!

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