The Virial Theorem, Part II: We Are Not Heated By Shrinkage

The Virial Theorem states that half of a gravitationally-bound system's potential energy goes into kinetic energy. It turns out that for a cloud of gas, the other half of $U$ goes into thermal radiation. We  know that the Sun started from the gravitational collapse of a giant cloud of gas. Let's hypothesize that the Sun is powered solely by this gravitational contraction, as was once posited by astronomers long ago. As it shrinks, its internal thermal energy increases, increasing its temperature and thereby causing it to radiate. How long would the Sun last if it was thermally radiating its current power output, $L_\odot = 4 \times 10^{33}$ erg s$^{- 1}$? This is known as the Kelvin-Helmholtz timescale. How does this timescale compare to the age of the oldest Moon rocks (about 4.5 billion years, also known as Gyr)?

The Kelvin-Helmholtz mechanism made an appearance in the last post, where we discussed how it played a role in the heating of exomoons, potentially rendering them uninhabitable. Here we're applying it to our star to see whether, at least when it comes to the age of the Sun, it's a viable way for the Sun to be producing its energy.

If we say that the Sun's thermal emission $E_T$ is equal half of the Sun's potential energy, we can (thanks to the legwork we did in the first post on the Virial Theorem) state that \[T_E = \frac{3GM_\odot^2}{10R_\odot}\] Plugging in the relevant constants gives us \[\frac{3(7 \times 10^{-8}) (2 \times 10^33)^2}{10(7\times 10^{10})} = 1.2 \times 10^{48} \: \: \textrm{erg}\] Now we'll assume that the Sun has been constantly producing its current energy output, $L_\odot = 4 \times 10^{33}$ erg s$^{-1}$. Dividing $E_T$ by this value gives us how long the Sun would last if it were only capable of producing energy via the Kelvin-Helmholtz mechanism.\[\frac{1.2 \times 10^{48} \: \: \textrm{erg}}{4 \times 10^{33} \: \: \textrm{erg s}^{-1}} = 3 \times 10^{14} \: \: \textrm{seconds}\]Converting this to years gives us about $9 \times 10^6$, or 9 million, years. That may seem like a long time, but compared to real geological or cosmological time, it's nothing. If Moon rocks date back 4.5 billion years, then the Sun must too--so something else is powering our solar system.

Not currently shrinking (source).