= Solution
For uniform density, <Kelvin-Helmholtz contraction> releases binding energy $|U|=3GM^2/(5R)$. If mass and luminosity are constant and stellar heating is negligible at 90 au, the contraction age is
$$
\boxed{t_{\rm KH}\simeq\frac{3GM^2}{5RL}}.
$$
Energy conservation, $L=-dU/dt$, gives
$$
\boxed{\dot R=-\frac{5LR^2}{3GM^2}}.
$$
The <virial theorem> places roughly half of the released gravitational energy into internal heat. Including that effect gives $t_{\rm KH}\simeq3GM^2/(10RL)$ and $\dot R\simeq-10LR^2/(3GM^2)$.
Back to article page
