= Solution
At $10\,\mathrm{au}$ from a four-solar-luminosity star, the incident flux is proportional to $4/10^2=1/25$, approximately the same as for <Jupiter> at $5\,\mathrm{au}$ around the Sun. Thus <Jupiter> supplies a possible old comparison at roughly the same mass and irradiation.
Assume the radius law holds from the young epoch to an old age of $4.5\,\mathrm{Gyr}$, that the irradiation history can be represented by the same fixed value, and that the present old radius is $R_J$. If $t$ is approximated by stellar age, then
$$
\frac{2R_J}{R_J}=\left(\frac{5\,\mathrm{Myr}}{4500\,\mathrm{Myr}}\right)^{-\alpha},
\qquad
\boxed{\alpha=\frac{\log2}{\log900}\simeq0.102.}
$$
If $t$ instead starts at completion of formation, the model's formation delay of up to $3\,\mathrm{Myr}$ leaves the young planet with thermal age between $2$ and $5\,\mathrm{Myr}$. Taking the old thermal age as approximately $4500\,\mathrm{Myr}$ then gives $\alpha\simeq0.090$–$0.102$.
Thus \b[an exponent of order 0.1] is a reasonable conditional estimate. A single radius at one age cannot determine both the exponent and normalization of a power law; the old <Jupiter> comparison and the assumptions about formation time and irradiation are essential.
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