Orbital Periods and Equilibrium Temperatures from Single TESS Transits with a Physics-Informed Neural Network

Planets with orbital periods longer than a TESS sector produce a single transit, and no periodogram method can measure their period: with one transit, every trial period longer than the observing baseline fits the data identically. We show this is not a sensitivity limit but a structural one -- across 16 confirmed single-transit planets, Box Least Squares returns a power spectrum that is numerically constant over 95% of its search grid, and widening that grid from 27 to 200 days changes the median error by-0.0 percentage points. Orbital period can instead be recovered from transit duration through Kepler's third law and transit geometry, requiring no search grid. A direct inversion of these equations underestimates the period in 14 of 15 targets with a median signed error of-69%, because assuming a central transit returns the shortest period consistent with an observed duration. A neural network trained to marginalise over the unobserved geometry removes this bias, reaching a median absolute error of 40.5% against Box Least Squares' 79.5%, with the true period inside the 1-sigma interval for 14 of 16 targets. For NGTS-38 b, whose 180.5 d period lies ten times beyond the longest contiguous span of its TESS sector, marginalising over the unobserved geometry recovers a posterior median of 190.7 d with the true period at the 47.8th percentile, against 18.5 d from Box Least Squares. Because equilibrium temperature scales as P(-1/3), the resulting factor-8.8 period interval compresses to a factor-2.1 temperature interval: T eq = 445 K with a 68% interval of 272-562 K, sufficient to place the planet relative to the habitable zone from a single observation.