# How did physicist calculate times like $10^{1500}$ and $10^{10^{76}}$ years?

If protons don't decay, iron stars are expected to form via quantum tunneling after $$10^{1500}$$ years, and they are expected to all have become neutron stars or black holes after $$10^{10^{26}}$$ to $$10^{10^{76}}$$ years. How did physicists calculate these insane values?

Let $$m,\,m_p$$ denote the electron and proton masses, and $$M\approx\frac14Am_p$$ the reduced mass of two $$\frac12A$$-nucleon nuclei that fuse to form $${}_{26}^{56}\text{Fe}$$. In terms of the fine structure constant $$\alpha=\frac{e^2}{4\pi\varepsilon c\hbar}$$ and a characteristic length scale $$d=Z^{-1/3}\alpha^{-1}\frac{\hbar}{mc}$$,\begin{align}U&=\frac{Z^2\alpha c\hbar}{4d}\\&=\frac{Z^{7/3}\alpha^2mc^2}{4},\\\sqrt{2MU}&=A^{1/2}Z^{7/6}\alpha c\sqrt{\frac{mm_p}{8}},\\S&=\frac{2d}{\hbar}\sqrt{2MU}\\&=A^{1/2}Z^{5/6}\sqrt{\frac{m_p}{2m}},\\e^S&\approx10^{1487.46}.\end{align}
For a radius-$$R$$ iron star of $$N$$ electrons, $$M=2Nm_p$$ and $$U=\frac{N^{5/3}\hbar^2}{2mx^2}$$ in the non-relativistic regime, so$$S=N^{4/3}\sqrt{\frac{8m_p}{m}}\ln\frac{R}{R_0},$$where $$R_0$$ is the radius at which relativity becomes relevant. If the logarithm is $$1$$,$$N=10^{56}\implies S\approx10^{2.44\times10^{76}}.$$