2026ApJ...997..167S

The irradiance received by a spherical body or a planet close to a spherically symmetric source does not follow the point-source approximation and the inverse-square variation of irradiation if spherical symmetry is broken. In the penumbral zones of the planet, spherical symmetry of the star reduces to an axial symmetry. Our work aims to put forward a fundamental explanation, using energy conservation, to determine the variation of irradiance in the penumbral zone on a close-in planet where the point-source approximation fails. Consequently, we propose a numerical model that accurately predicts the irradiance within the boundaries of the penumbral zone and the fully illuminated zone. Our analysis also corrects a previous study on exoplanet irradiation that violates energy conservation. We find that night-side illumination partially explains the observed night-side temperatures on the planets considered; this reduces reliance on heat transport models to explain the night-side temperature for the few exemplar rocky close-in planets, namely K2-141 b, 55 Cancri e, TOI-561 b, TOI-431 b, and Kepler-10 b, that are discussed in this work. We provide improved day-night contrast temperatures, considering an airless scenario, and highlight the need for revisiting the heat transport models associated with atmospheric modelling of planets where the night-side illumination is significant.