On the importance of geometry in exoplanet irradiation : Implications for the day-night contrast

TL;DR

This paper develops an energy-conserving, geometry-based framework to compute exoplanet irradiation when the star cannot be treated as a point source, showing that the inverse-square law breaks down in penumbral regions. Using a solid-angle approach and Poynting flux conservation, it derives when the IS law holds and when axial symmetry requires numerical treatment, and it provides InstellCa, a Python tool that computes irradiance across latitudes with high accuracy. The work reveals that night-side illumination can set a nonzero baseline temperature, influencing day-night contrasts and potentially reducing the need for strong atmospheric heat transport in airless or tenuous-atmosphere planets, with notable implications for planets like 55 Cancri e and K2-141 b. These findings offer a principled way to reinterpret thermal phase curves, refine atmospheric models, and prepare for JWST observations, while highlighting the importance of terminator geometry for exoplanet climates and potential habitability assessments.

Abstract

The irradiance received by a spherical body or a planet close to a spherically symmetric source does not follow the point-sized 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-sized 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.

Figures (6)

• Figure 1: Schematic representation of the Gaussian 2D surface constructed in subsection \ref{['mathodangle']}. An infinitesimal solid angle $\delta \Omega$ is centred in the centre of the star and reaches the surface of the planet. Along its path, it "cuts out" a portion of space between the star and the planet whose boundary is the surface under consideration. The arrows in the picture are the normal unit vectors to the surface. The yellow star denotes the star as equivalent to a point-sized source in case of spherical symmetry.
• Figure 2: Fig. (a) shows geometry of the common tangents to the stellar and planetary circles. The blue region is the set of all those positions from which the star is not entirely visible. An observer (represented in our case by the eye) sitting on an arbitrary blue point would see a partially eclipsed star and hence a non-zero tangential component of the flux. This opens the door to possible violations of the inverse-square law on the dark green region of the planet. However, one should note that the violations are only expected where the flux from the star is non-zero. For instance, in the black region on the planet beyond the exterior common tangents, the flux is naturally zero. Fig. (b) shows the observer seeing the complete star, which receives zero net tangential component of the flux due to polar symmetry, leaving S purely radial.
• Figure 3: An InstellCa plot showing irradiance received across the sub-stellar longitude of the exoplanets LHS 3844 b, 55 Cancri e, GJ 1252 b, and K2-141 b. The title shows the parameters used by the code (Table \ref{['Exo-parameters']}). We see largest manifestation of this effect for K2-141 b due to the larger visible angular size of the star ($\sim$ 25 degrees). For all planets, the numerical model overlaps with the inverse-square law for the fully illuminated zone. The slight difference between both approaches in the fully-illuminated zone is attributed to numerical error of less than one percent. In the penumbral zone, deviations are seen as the critical point of symmetry is crossed. The irradiance naturally falls to zero beyond the day-night terminator limit.
• Figure 4: Figure shows the derivative of the deviation from the IS law for the planet 55 Cancri e. The deviation from the IS law rapidly increases after the critical point of symmetry is crossed, implying consistency of the numerical results with the theoretical prediction. The red dotted line indicates the day-night terminator according to the points-sized source approximation, which is close to the poles.
• Figure 5: Variation of the predicted day-side temperature, which is concordant between our approach and the IS law, with multiple Bond albedo values. The observed brightness temperatures are taken from light curve analysis of each planet respectively hu2024secondaryzieba2022k2bonomo2025depthteske2025thick.