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

Question

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?

Answer 1

The key is to estimate tunnelling timescales by the WKB approximation. Your quoted figures are taken from Eqs. (41) and (47) in Dyson 1979 (this version is easier to search). While Eq. (41) follows from Eqs. (30) and (40) (the former summarizes usage of the aforementioned approximation), Eq. (47) follows from Eqs. (30) and (45).

First timescale

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 $d=Z^{-1/3}\alpha^{-1}\frac{\hbar}{mc}$,$$\begin{align}U&=\frac{Z^2\alpha c\hbar}{4d}\\&=\frac{Z^{7/3}\alpha^2mc^2}{4},\\\sqrt{2MU}&=A^{1/2}Z^{7/6}\alpha c\sqrt{\frac{mm_p}{8}},\\S&=\frac{2d}{\hbar}\sqrt{2MU}\\&=A^{1/2}Z^{5/6}\sqrt{\frac{m_p}{2m}},\\e^S&\approx10^{1487.46}.\end{align}$$

Second timescale

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 $1$,$$N=10^{56}\implies S\approx10^{2.44\times10^{76}}.$$