# What would be the maximum stable mass of an artificially created sphere of very cold, very pure hydrogen-1?

The Chandrasekhar limit assumes the white dwarf is made of carbon, but hydrogen has a much lower ratio of mass to electrons, so the limit should be different in that case. And electron degeneracy pressure might not even be the limiting factor, if neutronization or cosmic ray initiated fusion chain reactions are possible.

## 1 Answer

The "standard" Chandrasekhar mass is $$M_{\rm Ch} = 1.44\left(\frac{\mu_e}{2}\right)^{-2} M_{\odot}\ ,$$ where $$\mu_e$$ is the number of mass units per electron in the gas.

For pure hydrogen $$\mu_e= 1$$, so $$M_{\rm Ch}=5.76M_{\odot}.$$

At this mass, a Newtonian star supported by ideal electron degeneracy pressure would shrink to zero size and infinite density.

Would such an object ever exist? No, because it would become unstable due to General Relativity or Inverse Beta Decay or because nuclear reactions fuse the hydrogen into helium (with $$\mu_e = 2$$). All of the above will commence at finite densities and hence lower masses.

In the real universe (as opposed to a hypothetical one), you cannot assemble a cold hydrogen white dwarf without it being much hotter in its earlier life. We know that if such an object exceeds about $$0.08 M_{\odot}$$ (aka the brown dwarf limit), then eventually, its contracting core will commence thermonuclear hydrogen fusion.

So my real answer would be $$0.08M_{\odot}$$.

But you are going to ask, what if I could somehow extract all the potential energy and not allow it to heat the interior? Well, you would still get pyconuclear reactions caused by zero point energy and quantum tunneling in a solid proton lattice at around $$10^{9}$$ kg/m$$^3$$, even at cold temperatures. This will occur well before inverse beta decay, which would require electron Fermi (kinetic) energies of 0.78 MeV and hence densities of $$1.2\times 10^{10}$$ kg/m$$^3$$.

Back of the envelope: if we use non-relativistic degeneracy pressure, the mass radius relation of an ideal, cold white dwarf is $$R \simeq 0.013\left(\frac{\mu_e}{2}\right)^{-5/3} \left(\frac{M}{M_{\odot}}\right)^{-1/3} R_{\odot}$$ The density for $$\mu_e=1$$ is therefore $$\rho \simeq 2\times 10^7 \left(\frac{M}{M_{\odot}}\right)^2.$$

Thus a density of $$10^9$$ kg/m$$^3$$ will not be reached before the electrons become (mildly) relativistic. Unfortunately, that means running a numerical model to find the density at a given mass, which I can't do on my phone. However, roughly scaling the well-known equivalent model for a carbon white dwarf I would estimate that this density is reached at a hypothetical mass of $$\sim 0.5 M_{\rm Ch}=2.9M_{\odot}$$.