Star Cluster Within Self Schwarzschild Radius

[chill_factor]

I have a qualitative question to ask:

The Schwarzschild radius of matter is proportional to its mass.
The actual radius of the matter, assuming it is spherical, is proportional to the cube root of its mass.

This implies that the density required to form a Schwarzschild radius decreases as total mass increases.

Is it therefore conceivable that a massive star cluster could exist very near its own Schwarzschild radius, even though the stars themselves were still spaced at astronomical distances?

If this collection of stars were to be perturbed and sent into its own Schwarzschild radius (perhaps through gravitationally attracting a nearby gas cloud that pushes it over the limit), what would be a plausible mode of evolution for the system?

 

[mfb]

While I see no reason why this should be impossible in theory, the required star density or the size of this cluster would have to be extremely large. If you would compress the ~200 billion solar masses of the whole milky way to a black hole, it would have a radius of ~600 billion km, which is less than 0.1 light years. Long before you could reach this density, stellar collisions would happen in large amounts, making supernovae and big black holes everywhere.

If you go to the scale of the observable universe, it gets tricky due to dark energy, expansion and so on.

 

[Drakkith]

I don't believe there is any reasonable way for star formation to produce a star cluster that comes anywhere close to this.

 

[chill_factor]

I don't believe there is any reasonable way for star formation to produce a star cluster that comes anywhere close to this.

True. I was just curious as to what would happen as the radius of this mass approached itself schwarzschild radius. would there be any special effects?

If not a star system, then a diffuse gas cloud would also work. this would be similar to the gas clouds that existed in the early universe before star formation, and which could account for supermassive black holes existing.

 

[vociferous]

I find that the best way to answer these questions is to do the calculations yourself.

The Schwarzschild radius of M22 would be about 1.2 solar radii.

Now, keep in mind the the Schwarzschild radius scales linearly with mass as does the gravitational potential, which means that matter in hydrostatic equilibrium is not normally going to be anywhere near the Schwarzschild radius.

However, if it were to happen by some chance, there are two effects that would come into play.

The first is outward pressure created by gas pressure, as given by the ideal gas law (for really low masses, you can treat them as solids). This dominates in planets and some main sequence stars and is caused by the electromagnetic force.

The second is degeneracy pressure, which affects the central pressure in some large planets/brown dwarfs and stars.

The problem is, degeneracy pressure does not increase much with higher masses whereas central pressure increases at M^2 for non-relativistic cases. It is also the only thing keeping a mass from collapsing once the central pressure exceeds the gas pressure

The consequence of this is that, at the same proportional radius that a smaller mass will not collapse, a larger mass will, which means that if somehow you were able to pack anything close to the mass of a cluster into something near its Schwarzschild radius, there would likely be no stopping it from collapsing into its own Schwarzschild radius.

What happens inside the Schwarzschild radius, we cannot truly say, so there is no plausible evolution for that system that ever results in something we can learn about empirically. It becomes the playground of theoretical physics, especially those branches attempting to unite GR with Quantum mechanics.

If you look at the Jeans mass though, for large masses you essentially will be likely to get fragmentation, which is why the galaxy is not one giant black hole. If you have any molecular cloud massive enough that begins to collapse adiabatically, the only plausible evolution for that mass is to become either degenerate matter or a black hole, as it cannot fragment and it is too massive to be held up by gas pressure.

 

[lpetrich]

One can find where (G*M)/(R*c[/sup]2[/sup]) ~ 1 for some density rho ~ M/R³ (M = mass, R = radius).

That gives us G*rho*R²/c² ~ 1 or

R ~ c/sqrt(G*rho)
M ~ c³/sqrt(G³*rho)

So all one has to do is to find the average density of some kind of star cluster and plug it in.