Stable muon density inside a white dwarf star?

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.

Fun question. The muon density inside a white dwarf is negligible, because Fermi suppression does not really apply.

Fermi suppression is the technical name of the effect you were describing: the decrease in the rate of a process due to the fact that there are no free states to accommodate one of the decaying particles (an electron, in this case).

The muon has a mass of $105.6 MeV$, about 210 times more than the electron. Thus the electron resulting from the decay is necessarily highly relativistic. We shall then have Fermi suppression if the Fermi energy $E_F$ is larger than the energy the electron gains in the muon decay.

Now, the Fermi Energy for dense matter in the highly relativistic limit (see here, for instance) is $$ E_F = hc \left(\frac{3 n_e}{8\pi}\right)^{1/3} \approx 6\times 10^{-7} n_e^{1/3} eV $$

In order to have $E_F = 30 MeV$ (let us imagine that the other two particles carry away as much energy as the electron), we need $$ n_e \approx 10^{41}\; cm^{-3}\;\;\;. $$ Since inside a white dwarf there are as many protons as there are electrons, this imples a local density of $$ \rho \approx 10^{17}\; g \; cm^{-3} $$ which is high even for neutron stars, let alone white dwarves.

Thus the Fermi energy is never this ($30 MeV$) large in white dwarfs, and muon decays proceed unimpeded.