= Solution
Assume the planet and present-day <Jupiter> have the same <mass>, comparable composition and initial cooling normalization, and both can be described by the supplied <power-law planetary cooling> over the relevant ages. Take the age of present-day <Jupiter> to be $t_J\simeq4.5\,\mathrm{Gyr}$ and the young planet's age to be $0.050\,\mathrm{Gyr}$. The normalization cancels:
$$
\boxed{\frac{L_{\rm int}(50\,\mathrm{Myr})}{L_{\rm int,J}}
=\left(\frac{4.5}{0.050}\right)^{3/2}
=90^{3/2}\simeq8.5\times10^2.}
$$
The 5-AU orbit is used to regard stellar irradiation as modest compared with a hot-Jupiter orbit; the ratio is specifically intrinsic cooling, not the sum of intrinsic and reradiated <luminosity>. Initial <entropy> and irradiation can change the assumed normalization, so this estimate is not independent of formation conditions.
Such a young self-luminous giant is a favourable target for near-infrared <exoplanet direct imaging>. The increased intrinsic <luminosity> improves its contrast with the host star, and a wide physical orbit is more readily separated on the sky than a hot-Jupiter orbit. Two essential observing considerations are:
* \b[Angular separation:] at distance $d$ parsecs, a projected 5-AU separation is at most roughly $5/d$ arcseconds. It must exceed the instrumental inner working angle; orbital projection can make it smaller. Nearby targets and <adaptive optics> help.
* \b[Planet-star contrast and wavelength:] the band must balance the young planet's <thermal radiation>, stellar leakage, detector sensitivity and thermal background. Suppression with a <coronagraph> and stable calibration are required; a large intrinsic bolometric ratio to old <Jupiter> is not itself the measured contrast with the star.
Thus \b[direct imaging of a nearby young system, with sufficient angular resolution and contrast, is the natural discovery method], subject to the actual stellar brightness and instrumental limits.
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