= Solution
Build the star by adding spherical shells. With $m(r)$ the enclosed <mass>, its <gravitational binding energy> is
$$
\Omega=-\int_0^{M_\odot}\frac{Gm}{r(m)}\,dm.
$$
For a nondegenerate extended star, $r(m)<R_\odot$ on the interior, so
$$
\boxed{|\Omega|>\frac{G}{R_\odot}\int_0^{M_\odot}m\,dm
=\frac{GM_\odot^2}{2R_\odot}.}
$$
This is a <gravitational binding-energy lower bound for a spherical star>. The <Kelvin-Helmholtz cooling time> is of order
$$
\boxed{\tau_{\rm KH}\sim\frac{GM_\odot^2}{R_\odot L_\odot}\simeq3.1\times10^7\,\mathrm{yr}.}
$$
The displayed energy bound gives $|\Omega|/L_\odot>1.6\times10^7$ years; it supplies a lower bound on that reservoir-to-luminosity ratio. A stellar structure model sets its coefficient. For an ideal monatomic, gas-supported star the <virial theorem> puts about half the released gravitational energy into heat, leaving half available for radiation; a uniform-density model would give about $9\times10^6$ years. These contraction estimates are all of order $10^7$ years, much shorter than the nuclear lifetime.
Back to article page
