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Incoming links: Power-law planetary cooling
Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 3 a ii Solution Created 2026-10-03 Updated 2026-10-06
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 and the young planet's age to be . The normalization cancels: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:
- Angular separation: at distance parsecs, a projected 5-AU separation is at most roughly arcseconds. It must exceed the instrumental inner working angle; orbital projection can make it smaller. Nearby targets and adaptive optics help.
- 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 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.