 Solution

The conventional Kelvin-Helmholtz cooling time is of order $G{M}^2/(RL)$. For the monatomic uniform-density stellar model, its precise accessible total-energy reservoir is $|E|=3G{M}^2/(10R)$, soUsing $M_{\odot }\simeq 1.9885\times 10^{30}\mathrm{kg}$, $R_{\odot }\simeq 6.957\times 10^8\mathrm{m}$ and $L_{\odot }\simeq 3.828\times 10^{26}\mathrm{W}$, the usual order-of-magnitude normalization without the structural factor is $t_{\mathrm{KH}}\simeq 3.1\times 10^7\mathrm{yr}$. The full gravitational binding reservoir $|\Omega |/L$ is about $1.9\times 10^7\mathrm{yr}$, but that overcounts the radiatable energy in virial equilibrium.

The Sun and the Solar system are approximately $4.6\times 10^9\mathrm{yr}$ old, much older than any of these contraction estimates. Gravitational contraction cannot sustain the Sun's long-lived present power. The dominant long-term source is stellar nuclear fusion. Contraction can still supply transient luminosity, especially in a pre-main-sequence star. The timescale comparison rules out a contraction-only explanation over the observed age; it does not assert that all gravitational energy release is absent.