Frank Shu's astrophysics text "the Physical Universe" provides the formulas you need to calculate the size of a neutron star of any given mass
this is controlled by the degeneracy pressure, or fermion pressure, of neutrons
at a central density rho there is only so much fermion pressure and unless it is enough to balance the central pressure of a self gravitating sphere of given mass and size, the star will collapse further----until (and if) the central fermion pressure rises to where it equals the hydrostatic pressure at the center.
so one calculates the size of a neutron star of mass M by setting two formulas equal and solving for R.
It will turn out that MR
3 is a constant (I will tell you later this constant).
So if you could increase the mass of a neutron star by 8-fold, the volume would shrink 8-fold and the radius would shrink by half.
the more massive neutron stars are the smaller ones.
At some mass (perhaps 3 solar masses) the radius is less than the Schwarzschild radius----PRESTO: the neutron star is a black hole. It is all inside its event horizon. What things are like inside there we cannot yet say.
Do not worry about "singularity". this word refers to fact that GR breaks down at very high density. The GR theory reaches the limits of its applicability and fails to compute. When there is a quantum theory of general relativity then perhaps we humans will understand better about inside the Schwarzschild radius. But for now we can say "it is a black hole, now the star is all inside the event horizon."
THE HEISENBERG does not prevent a neutron star from being compressed to size of one cubic meter! All Heisenberg says is that when the Δx of the pointparticle is very small its momentum uncertainty Δp is very large!
If you have many neutrons or quarks in a box (a kind of fermion gas) and you compress it this increases the kinetic energy of the gas inside. So you are doing work! So there is pressure!
The Heisenberg just says how much pressure.
You can still keep squeezing the box, if you are strong enough.
They are only points. And Δx can be made very small.
I will get the approximate formula for the fermion pressure (in terms of the number density of neutrons----or equivalently in terms of mass density) and post it later.
The analysis is very similar to the way the Chandrasekhar mass formula is derived---but that concerns the fermi pressure of electrons being overcome, allowing collapse to neutron matter.
We next must consider the fermi pressure of the neutrons being overcome to the point that a black hole forms.
WHOAH! I had to be away from computer and I see now futz has answered this. My post is not really needed. If you want the formulas for fermion pressure and size of neutron star, just say. Otherwise I will forget it.
Originally posted by Dimitri Terryn
After reading "Hyperspace" by Kaku, I was slightly puzzled after his chapter on collapsing stars.
He states that white dwarfs and neutron stars remain stable, because the exclusion principle counter-acts the gravitational collapse.
He then sais the stars who have reached the end of the fusion cycle and who are composed of mostly iron have no forces balancing gravity anymore, and hence these collapse to a point.
My question is, where does the exclusion principle go here? Why would it suddenly be possible to precisely know the quantum state of the matter inside the singularity? Obviously I'm missing something here, but some insight you might would be greatly appreciated.
-Dimi
). I suspected it had something to do about us not being able to observe the thing once it had collapsed to a black hole. That and the fact that at those distances GR does indeed break down.