homework and exercises - Stable muon density inside a white dwarf star?

It occurs to me (though I'm hardly the first) that the decay $$ \mu^- \to e^- + \bar \nu_e + \nu_\mu $$ should be forbidden in electron-degenerate matter, since there must be an empty state available to receive the electron. How would one take this fact to make an order-of-magnitude estimate of the equilibrium muon density at the core of a white dwarf star?

(Technical answers and illuminating references are welcome.)

Answer

Though I agree with the logic of MariusMatutiae, I find I cannot reproduce their quantitative answer.

I get an electron number density of $1.2 \times 10^{41}$ m$^{-3}$ (is it just a unit thing?) for an electron Fermi energy of 30MeV.

In a carbon white dwarf with 2 mass units per electron, the Fermi energy of the electrons reaches 30MeV at densities of $4\times 10^{14}$ kg/m$^{3}$ - i.e. at densities which still do exceed the maximum possible in a White Dwarf because of instabilities caused by inverse beta decay or General Relativity (the maximum density of a WD is more like a few $10^{13}$ kg/m$^3$). But not as high as densities in a neutron star.

I have produced an applet that allows you to explore the parameter space in detail.

http://www.geogebratube.org/student/b87651#material/28528

OK, but even given that, where do the muons come from? Actually, you need to create them with an energy of 105.6MeV from electrons and anti-neutrinos. If they reach some kind of equilibrium then the chemical potential (the Fermi energy) of the electrons and muons should be equal. Thus the electron energy threshold for muon production is normally considered to be more like 105.6MeV and consequently a factor of $(100/30)^3$ higher electron number densities and mass densities are required.

A similar calculation shows that in neutron stars, muon production is not really viable until densities reach several $\times10^{17}$ kg/m$^3$. It is a much higher density here because (a) the electrons and muons have the same Fermi energies when they are at equilibrium (b) the number of mass units per electron is more like 60.