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I am trying to find real physical examples of (self-gravitating) astrophysical systems that are appropriately confined and can thus be seen as in equilibrium. Modelling-wise, you can theoretically put the system in a box and get stable systems in that way. The question is: are there real examples of astrophysical systems that are actually as if they are in a box? The role of the box could be played e.g. by an external potential, a potential well or by tidal effects.

I am thinking in particular about real examples of bounded globular clusters as they look like the best candidates.

Edit (2): All globular clusters are gravitationally bound in a sense, but I am looking for self-gravitating systems that are confined in the sense that we can consider the system to behave like it is a gas confined in a box. Otherwise, the system would have (i) evaporation and (ii) for self-gravitating systems the stars would end up clumping in the centre and expelling material outside, this would make the system unstable triggering gravothermal catastrophe. Putting the system in a theoretical box solves the problem and ensures the system can stay stable, this is clear from models but I'm looking for physical examples of potentials that can do the job at a high approximation.

(I'm roughly asking what user 'uhoh' says and Anders Sandberg's reply is the kind of example I'm looking for)

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    $\begingroup$ Unclear what you are asking for. All globulars are bound and all are suffering from evaporation. $\endgroup$
    – ProfRob
    Oct 20, 2023 at 16:00
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    $\begingroup$ Are you asking for n-body systems that are so tightly bound (so low energy) that none can escape to infinity even if all the others come to rest? If so (and I think that's what you might be asking) it's not yet completely clear to at least some readers. voting to leave open for a bit to give the OP some time to clarify. The close-and-reopen cycle takes many days and is not really productive in this case - I think this can be cleared up fairly quickly without all the drama. $\endgroup$
    – uhoh
    Oct 21, 2023 at 0:56
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    $\begingroup$ Would a cluster of normal stars at the centre of a big dark matter halo count? The halo establishes a deep and stable potential well with a high escape velocity that makes evaporation fail. $\endgroup$ Oct 21, 2023 at 12:04
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    $\begingroup$ @uhoh this is roughly what I mean, I edited and clarified, it's kind of a narrow question. $\endgroup$ Oct 21, 2023 at 14:54
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    $\begingroup$ @AndersSandberg that is the kind of example I'm looking for, although I was wondering if the dark matter halo will end up attracting the stars towards the outer part of the cluster due to the dark matter's gravity $\endgroup$ Oct 21, 2023 at 14:55

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You are asking for a star cluster that sits at the bottom of an infinite potential well. Such clusters do not exist because they are largely the source of the potential that they reside in.

There are a population of objects in the universe known as dwarf spheroidal galaxies. These look a bit like globular clusters but appear to be embedded in dark matter. An example is "Segue 2" (Kirby et al. 2013). But even here, all that happens is that the stars pick up extra kinetic energy as they "thermalize" in the dark matter potential. They will still have a tail of high kinetic energy objects that can escape from the system. The virial theorem says that the kinetic energy and gravitational potential will always be proportional if the system reaches equilibrium.

Another possibility is then to look for star clusters that are out of equilibrium and sitting at the bottom of a deep potential well. A possibility there is to look at very young (star forming) clusters that are embedded in lots of gas. The gas provides a deep potential well and, if young enough, the stars have not "virialised".

Most well-known young star clusters are not embedded - otherwise we wouldn't see them and take pretty pictures. But they can be surveyed by infrared telescopes - see for example the review by Ascenso (2018). Possibly what you are looking for might be provided by something like Westerlund 2.

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  • $\begingroup$ What a "cool" answer! You've inspired a follow-up: If a cluster of stars in dynamical equilibrium falls into a much larger blob of dark matter, will it get hotter and expand? Will it stop? $\endgroup$
    – uhoh
    Oct 22, 2023 at 11:27
  • $\begingroup$ Thanks, this is what I was looking for. I'm wondering if the first kind of system you mention can be really in thermodynamical equilibrium at some point but I shall ask that in a further thread. $\endgroup$ Oct 23, 2023 at 15:30
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    $\begingroup$ @FriedBarking you can;t really say it's in thermodynamical equilibrium because stars will be evaporating. The relaxation timescale is $\sim 10^{11}(v/10{\rm km/s})^3(m/M_\odot)^{-2} (n/{\rm 1 pc}^{-3})^{-1}$ in years. $\endgroup$
    – ProfRob
    Oct 23, 2023 at 16:11
  • $\begingroup$ Yes, indeed my original question was mainly concerned with the possibility of systems placed in a potential well so deep that the evaporation rate is so low that they are effectively in thermodynamical equilibrium (or at least in a sufficiently stable metastable state). Talking about thermodynamic equilibrium also core collapse seems to play a role and I wonder if having a small size strongly confined cluster can help avoid core collapse and support a form of approximate thermodynamic equilibrium. But I am aware that all these issues are outside the scope of the main question of the thread $\endgroup$ Oct 23, 2023 at 18:31
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Disregarding the example of globular clusters (which are all undergoing evaporation, as ProfRob points out), there are a couple other candidates for "isolated" astrophysical systems.

The first is void galaxies, which are located in regions of particularly empty space, and rarely (if ever) interact with other galaxies. These galaxies are effectively "systems in a box", although I believe they might still interact with the intergalactic medium in meaningful ways.

The other example I thought of is a bit crazier; assuming that the Universe will eventually undergo heat death (i.e. that cosmic expansion continues to accelerate), eventually galaxies will be accelerated away from each other that they will no longer be able to interact. At this point in the Universe, these galaxies will all be "systems in a box".

To digress, practically speaking, there are globular clusters with relaxation times that are much larger than the age of the Universe, and can be expected to remain bound for extremely large periods of time. Similarly, the evaporation times of Milky Way-like galaxies are on the orders of tens of trillions of years, so these objects can be treated as evaporation-free without much loss of precision. As $t\rightarrow\infty$ I think these objects would technically be expected to evaporate, but by then I think we expect stars to no longer be fusing and galaxies to start being flung apart by cosmic expansion, so I'm not sure that a concept like "evaporation time" matters at that point.

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  • $\begingroup$ I'd like to bounty both answer so I'll add a 2nd bounty. The SE algorithm requires each succssive bounty to double, so when there's two good answers I give the larger bounty to the lower rep user. Thanks! $\endgroup$
    – uhoh
    Oct 28, 2023 at 0:55
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Well, things like small asteroids are self-gravitating and confined to a pretty well defined volume. We believe they are approximately in thermal equilibrium. But it isn't gravity that brings them to equilibrium: it's molecular forces.

Larger bodies like planets and stars are generally not in thermal equilibrium: they are hotter on the inside than the outside.

For bigger things where gravity dominates the interaction, things get stranger. From kiloparsec to megaparsec scale we observe a natural velocity scale of approximately 0.1% of the speed of light. Velocity dispersions of stars within galaxies, galaxies within clusters, and clusters within superclusters are all similar. This is not Boltzmann's energy equipartition: the velocity dispersion is independent of mass. Gravitational interactions apparently don't tend toward equipartion.

Dimensional analysis suggests that this is related to the observed fractal dimension of gravitational clustering of ~1 (Mandelbrot: "Fractals: Form Chance and Dimension" chapter V). But, as with many observed fractal phenomena, the usual reductionism of theoretical physics seems difficult to apply to this phenomenon.

So, above the scale of asteroids, we just don't see systems in a state resembling classical thermal equilibrium.

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  • $\begingroup$ This is really interesting to think about! Considering For "Rubble pile" objects made of macroscopic chunks of stuff above a certain size, it's got to be gravity that holds them together; it's not exactly a squared/cubed thing but close. But yes for objects made of smaller bits that include binders like ice and dust, an der Waals forces will contribute and certainly in some cases dominate the binding energy - until you bump into them with something. $\endgroup$
    – uhoh
    Oct 29, 2023 at 0:07

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