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Suppose that in the intergalactic space far from any significant gravitational attractors there is a relatively small concentration of He-4 atoms. Due to gravitational attraction fermions in this case would form a sphere (gas planet), but as bosons He-4 atoms aren't affected by exclusion prinicple, so what's holding them back from collapsing into singularity?

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Some related questions: physics.stackexchange.com/questions/22049/… physics.stackexchange.com/questions/90405/… and I believe there is at least one more, but I haven't found it. –  dmckee Apr 22 at 19:20

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Helium nuclei behave like bosons only in phenomena where their integrity is preserved and they can be assumed as point-like particles.

When you compress them a lot this is not the case any more. Single protons and neutrons will start to interact with each other and they are fermions. You have no chance to condensate them in a black hole, indeed what you get are triple-alpha processes in which helium is fused to $^{12}$C.

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Well, you get a chance if there are enough of them in a small enough area, which is really hard, but besides that part yes. –  trysis Apr 22 at 17:18
    
@trysis You are theoretically correct, but due to the radiation pressure we do not know any case in which that could happen avoiding to burn everything up to iron. –  DarioP Apr 22 at 17:28
    
Technically, that's not true: see #10 in Ten things you don't know about black holes by Phil Plait. A solar-system sized ball of helium at one atmosphere, i.e., well below the density necessary to fuse, would be a black hole. –  Harry Johnston Apr 22 at 21:34
    
@HarryJohnston: What wizardry are you using to create a solar-system sized ball of helium, and maintain it uniformly at one atmosphere? –  Iain Galloway Apr 23 at 11:34
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@Harry Johnston: Seems likely to me that you'd end up with a burning star long before you'd end up with a black hole cf. every star in the sky :D –  Iain Galloway Apr 24 at 8:15

If it is a planetary size mass, then thermal pressure from the kinetic energy of the atoms would prevent collapse.

For larger masses, helium fusion would create radiation pressure to stop the collapse.

Also, helium plasma would contain free electrons which would cause electron degenarcy pressure. Red giants of less than 2 solar masses initially have helium cores which are sustained by electron degeneracy pressure, until temperature is hot enough for helium fusion to start.

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