= Solution
The conventional <Kelvin-Helmholtz cooling time> is of order $GM^2/(RL)$. For the monatomic <uniform-density stellar model>, its precise accessible total-energy reservoir is $|E|=3GM^2/(10R)$, so
$$
\boxed{t_{\rm model}=\frac{|E|}{L}=\frac{3GM_\odot^2}{10R_\odot L_\odot}\simeq9.4\times10^6\,\mathrm{yr}.}
$$
Using $M_\odot\simeq1.9885\times10^{30}\,\mathrm{kg}$, $R_\odot\simeq6.957\times10^8\,\mathrm m$ and $L_\odot\simeq3.828\times10^{26}\,\mathrm W$, the usual order-of-magnitude normalization without the structural factor is $t_{\rm KH}\simeq3.1\times10^7\,\mathrm{yr}$. The full gravitational binding reservoir $|\Omega|/L$ is about $1.9\times10^7\,\mathrm{yr}$, but that overcounts the radiatable energy in virial equilibrium.
The <Sun> and the <Solar system> are approximately $4.6\times10^9\,\mathrm{yr}$ old, much older than any of these contraction estimates. \b[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.
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